# Real 2×2 Eigenvalue and Nonzero Eigenvector Foundations

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This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.

batch-20261006-advance-20261007-r04-eigenvalue-foundations · 2026-10-06 · 100 questions

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.



## 1. Eigenpair Candidate with a Nonzero Vector

Over R, the 2×2 matrix is M=[[2, 1], [0, 3]]. Is λ=99, v=[0, 0] a real eigenpair under the nonzero-eigenvector convention?

View answer and explanationNo

Mv=[0, 0] and λv=[0, 0]. Both equality and v≠0 are required, giving No. The zero vector is not an eigenvector, even though the equality then holds; zero or negative eigenvalues are not automatically excluded.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r04-eigenvalue-0001/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-eigenpair-candidate · candidate





## 2. Eigenpair Candidate with a Nonzero Vector

Over R, the 2×2 matrix is M=[[-11, 0], [0, 3]]. Is λ=3, v=[0, 1] a real eigenpair under the nonzero-eigenvector convention?

View answer and explanationYes

Mv=[0, 3] and λv=[0, 3]. Both equality and v≠0 are required, giving Yes. The zero vector is not an eigenvector, even though the equality then holds; zero or negative eigenvalues are not automatically excluded.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r04-eigenvalue-0002/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-eigenpair-candidate · candidate





## 3. Eigenpair Candidate with a Nonzero Vector

Over R, the 2×2 matrix is M=[[-10, 0], [0, 4]]. Is λ=-10, v=[1, 1] a real eigenpair under the nonzero-eigenvector convention?

View answer and explanationNo

Mv=[-10, 4] and λv=[-10, -10]. Both equality and v≠0 are required, giving No. The zero vector is not an eigenvector, even though the equality then holds; zero or negative eigenvalues are not automatically excluded.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r04-eigenvalue-0003/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-eigenpair-candidate · candidate





## 4. Eigenpair Candidate with a Nonzero Vector

Over R, the 2×2 matrix is M=[[-9, 0], [0, 5]]. Is λ=-8, v=[1, -1] a real eigenpair under the nonzero-eigenvector convention?

View answer and explanationNo

Mv=[-9, -5] and λv=[-8, 8]. Both equality and v≠0 are required, giving No. The zero vector is not an eigenvector, even though the equality then holds; zero or negative eigenvalues are not automatically excluded.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r04-eigenvalue-0004/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-eigenpair-candidate · candidate





## 5. Eigenpair Candidate with a Nonzero Vector

Over R, the 2×2 matrix is M=[[4, 1], [1, 4]]. Is λ=5, v=[1, 1] a real eigenpair under the nonzero-eigenvector convention?

View answer and explanationYes

Mv=[5, 5] and λv=[5, 5]. Both equality and v≠0 are required, giving Yes. The zero vector is not an eigenvector, even though the equality then holds; zero or negative eigenvalues are not automatically excluded.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r04-eigenvalue-0005/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-eigenpair-candidate · candidate





## 6. Eigenpair Candidate with a Nonzero Vector

Over R, the 2×2 matrix is M=[[5, 0], [1, 6]]. Is λ=5, v=[1, -1] a real eigenpair under the nonzero-eigenvector convention?

View answer and explanationYes

Mv=[5, -5] and λv=[5, -5]. Both equality and v≠0 are required, giving Yes. The zero vector is not an eigenvector, even though the equality then holds; zero or negative eigenvalues are not automatically excluded.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r04-eigenvalue-0006/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-eigenpair-candidate · candidate





## 7. Eigenpair Candidate with a Nonzero Vector

Over R, the 2×2 matrix is M=[[-6, 0], [0, 8]]. Is λ=-6, v=[1, 1] a real eigenpair under the nonzero-eigenvector convention?

View answer and explanationNo

Mv=[-6, 8] and λv=[-6, -6]. Both equality and v≠0 are required, giving No. The zero vector is not an eigenvector, even though the equality then holds; zero or negative eigenvalues are not automatically excluded.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r04-eigenvalue-0007/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-eigenpair-candidate · candidate





## 8. Eigenpair Candidate with a Nonzero Vector

Over R, the 2×2 matrix is M=[[-5, 0], [0, 9]]. Is λ=-4, v=[1, -1] a real eigenpair under the nonzero-eigenvector convention?

View answer and explanationNo

Mv=[-5, -9] and λv=[-4, 4]. Both equality and v≠0 are required, giving No. The zero vector is not an eigenvector, even though the equality then holds; zero or negative eigenvalues are not automatically excluded.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r04-eigenvalue-0008/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-eigenpair-candidate · candidate





## 9. Eigenpair Candidate with a Nonzero Vector

Over R, the 2×2 matrix is M=[[-4, 0], [0, 10]]. Is λ=-4, v=[1, 0] a real eigenpair under the nonzero-eigenvector convention?

View answer and explanationYes

Mv=[-4, 0] and λv=[-4, 0]. Both equality and v≠0 are required, giving Yes. The zero vector is not an eigenvector, even though the equality then holds; zero or negative eigenvalues are not automatically excluded.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r04-eigenvalue-0009/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-eigenpair-candidate · candidate





## 10. Eigenpair Candidate with a Nonzero Vector

Over R, the 2×2 matrix is M=[[-3, 0], [0, 11]]. Is λ=11, v=[0, 1] a real eigenpair under the nonzero-eigenvector convention?

View answer and explanationYes

Mv=[0, 11] and λv=[0, 11]. Both equality and v≠0 are required, giving Yes. The zero vector is not an eigenvector, even though the equality then holds; zero or negative eigenvalues are not automatically excluded.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r04-eigenvalue-0010/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-eigenpair-candidate · candidate





## 11. Eigenpair Candidate with a Nonzero Vector

Over R, the 2×2 matrix is M=[[-2, 0], [0, 12]]. Is λ=-2, v=[1, 1] a real eigenpair under the nonzero-eigenvector convention?

View answer and explanationNo

Mv=[-2, 12] and λv=[-2, -2]. Both equality and v≠0 are required, giving No. The zero vector is not an eigenvector, even though the equality then holds; zero or negative eigenvalues are not automatically excluded.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r04-eigenvalue-0011/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-eigenpair-candidate · candidate





## 12. Eigenpair Candidate with a Nonzero Vector

Over R, the 2×2 matrix is M=[[-1, 0], [0, 13]]. Is λ=0, v=[1, -1] a real eigenpair under the nonzero-eigenvector convention?

View answer and explanationNo

Mv=[-1, -13] and λv=[0, 0]. Both equality and v≠0 are required, giving No. The zero vector is not an eigenvector, even though the equality then holds; zero or negative eigenvalues are not automatically excluded.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r04-eigenvalue-0012/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-eigenpair-candidate · candidate





## 13. Eigenpair Candidate with a Nonzero Vector

Over R, the 2×2 matrix is M=[[0, 0], [0, 14]]. Is λ=0, v=[1, 0] a real eigenpair under the nonzero-eigenvector convention?

View answer and explanationYes

Mv=[0, 0] and λv=[0, 0]. Both equality and v≠0 are required, giving Yes. The zero vector is not an eigenvector, even though the equality then holds; zero or negative eigenvalues are not automatically excluded.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r04-eigenvalue-0013/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-eigenpair-candidate · candidate





## 14. Eigenpair Candidate with a Nonzero Vector

Over R, the 2×2 matrix is M=[[1, 0], [0, 15]]. Is λ=15, v=[0, 1] a real eigenpair under the nonzero-eigenvector convention?

View answer and explanationYes

Mv=[0, 15] and λv=[0, 15]. Both equality and v≠0 are required, giving Yes. The zero vector is not an eigenvector, even though the equality then holds; zero or negative eigenvalues are not automatically excluded.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r04-eigenvalue-0014/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-eigenpair-candidate · candidate





## 15. Eigenpair Candidate with a Nonzero Vector

Over R, the 2×2 matrix is M=[[2, 0], [0, 16]]. Is λ=2, v=[1, 1] a real eigenpair under the nonzero-eigenvector convention?

View answer and explanationNo

Mv=[2, 16] and λv=[2, 2]. Both equality and v≠0 are required, giving No. The zero vector is not an eigenvector, even though the equality then holds; zero or negative eigenvalues are not automatically excluded.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r04-eigenvalue-0015/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-eigenpair-candidate · candidate





## 16. Eigenpair Candidate with a Nonzero Vector

Over R, the 2×2 matrix is M=[[3, 0], [0, 17]]. Is λ=4, v=[1, -1] a real eigenpair under the nonzero-eigenvector convention?

View answer and explanationNo

Mv=[3, -17] and λv=[4, -4]. Both equality and v≠0 are required, giving No. The zero vector is not an eigenvector, even though the equality then holds; zero or negative eigenvalues are not automatically excluded.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r04-eigenvalue-0016/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-eigenpair-candidate · candidate





## 17. Eigenpair Candidate with a Nonzero Vector

Over R, the 2×2 matrix is M=[[4, 0], [0, 18]]. Is λ=4, v=[1, 0] a real eigenpair under the nonzero-eigenvector convention?

View answer and explanationYes

Mv=[4, 0] and λv=[4, 0]. Both equality and v≠0 are required, giving Yes. The zero vector is not an eigenvector, even though the equality then holds; zero or negative eigenvalues are not automatically excluded.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r04-eigenvalue-0017/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-eigenpair-candidate · candidate





## 18. Eigenpair Candidate with a Nonzero Vector

Over R, the 2×2 matrix is M=[[5, 0], [0, 19]]. Is λ=19, v=[0, 1] a real eigenpair under the nonzero-eigenvector convention?

View answer and explanationYes

Mv=[0, 19] and λv=[0, 19]. Both equality and v≠0 are required, giving Yes. The zero vector is not an eigenvector, even though the equality then holds; zero or negative eigenvalues are not automatically excluded.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r04-eigenvalue-0018/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-eigenpair-candidate · candidate





## 19. Eigenpair Candidate with a Nonzero Vector

Over R, the 2×2 matrix is M=[[6, 0], [0, 20]]. Is λ=6, v=[1, 1] a real eigenpair under the nonzero-eigenvector convention?

View answer and explanationNo

Mv=[6, 20] and λv=[6, 6]. Both equality and v≠0 are required, giving No. The zero vector is not an eigenvector, even though the equality then holds; zero or negative eigenvalues are not automatically excluded.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r04-eigenvalue-0019/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-eigenpair-candidate · candidate





## 20. Eigenpair Candidate with a Nonzero Vector

Over R, the 2×2 matrix is M=[[7, 0], [0, 21]]. Is λ=8, v=[1, -1] a real eigenpair under the nonzero-eigenvector convention?

View answer and explanationNo

Mv=[7, -21] and λv=[8, -8]. Both equality and v≠0 are required, giving No. The zero vector is not an eigenvector, even though the equality then holds; zero or negative eigenvalues are not automatically excluded.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r04-eigenvalue-0020/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-eigenpair-candidate · candidate





## 21. Eigenpair Candidate with a Nonzero Vector

Over R, the 2×2 matrix is M=[[8, 0], [0, 22]]. Is λ=8, v=[1, 0] a real eigenpair under the nonzero-eigenvector convention?

View answer and explanationYes

Mv=[8, 0] and λv=[8, 0]. Both equality and v≠0 are required, giving Yes. The zero vector is not an eigenvector, even though the equality then holds; zero or negative eigenvalues are not automatically excluded.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r04-eigenvalue-0021/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-eigenpair-candidate · candidate





## 22. Eigenpair Candidate with a Nonzero Vector

Over R, the 2×2 matrix is M=[[9, 0], [0, 23]]. Is λ=23, v=[0, 1] a real eigenpair under the nonzero-eigenvector convention?

View answer and explanationYes

Mv=[0, 23] and λv=[0, 23]. Both equality and v≠0 are required, giving Yes. The zero vector is not an eigenvector, even though the equality then holds; zero or negative eigenvalues are not automatically excluded.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r04-eigenvalue-0022/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-eigenpair-candidate · candidate





## 23. Eigenpair Candidate with a Nonzero Vector

Over R, the 2×2 matrix is M=[[10, 0], [0, 24]]. Is λ=10, v=[1, 1] a real eigenpair under the nonzero-eigenvector convention?

View answer and explanationNo

Mv=[10, 24] and λv=[10, 10]. Both equality and v≠0 are required, giving No. The zero vector is not an eigenvector, even though the equality then holds; zero or negative eigenvalues are not automatically excluded.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r04-eigenvalue-0023/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-eigenpair-candidate · candidate





## 24. Eigenpair Candidate with a Nonzero Vector

Over R, the 2×2 matrix is M=[[11, 0], [0, 25]]. Is λ=12, v=[1, -1] a real eigenpair under the nonzero-eigenvector convention?

View answer and explanationNo

Mv=[11, -25] and λv=[12, -12]. Both equality and v≠0 are required, giving No. The zero vector is not an eigenvector, even though the equality then holds; zero or negative eigenvalues are not automatically excluded.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r04-eigenvalue-0024/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-eigenpair-candidate · candidate





## 25. Eigenpair Candidate with a Nonzero Vector

Over R, the 2×2 matrix is M=[[12, 0], [0, 26]]. Is λ=12, v=[1, 0] a real eigenpair under the nonzero-eigenvector convention?

View answer and explanationYes

Mv=[12, 0] and λv=[12, 0]. Both equality and v≠0 are required, giving Yes. The zero vector is not an eigenvector, even though the equality then holds; zero or negative eigenvalues are not automatically excluded.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r04-eigenvalue-0025/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-eigenpair-candidate · candidate





## 26. Distinct Real Eigenvalues of a Diagonal Matrix

Over R, the 2×2 matrix is diag(0,0). Give its distinct real eigenvalues as an increasing set with each value once.

View answer and explanation{0}

The two diagonal values give {0}. Equal values are listed once; this is a set of distinct eigenvalues, not an algebraic-multiplicity count.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r04-eigenvalue-0026/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-eigenvalue-diagonal-set · candidate





## 27. Distinct Real Eigenvalues of a Diagonal Matrix

Over R, the 2×2 matrix is diag(-2,-2). Give its distinct real eigenvalues as an increasing set with each value once.

View answer and explanation{-2}

The two diagonal values give {-2}. Equal values are listed once; this is a set of distinct eigenvalues, not an algebraic-multiplicity count.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r04-eigenvalue-0027/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-eigenvalue-diagonal-set · candidate





## 28. Distinct Real Eigenvalues of a Diagonal Matrix

Over R, the 2×2 matrix is diag(-10,3). Give its distinct real eigenvalues as an increasing set with each value once.

View answer and explanation{-10, 3}

The two diagonal values give {-10, 3}. Equal values are listed once; this is a set of distinct eigenvalues, not an algebraic-multiplicity count.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r04-eigenvalue-0028/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-eigenvalue-diagonal-set · candidate





## 29. Distinct Real Eigenvalues of a Diagonal Matrix

Over R, the 2×2 matrix is diag(-9,4). Give its distinct real eigenvalues as an increasing set with each value once.

View answer and explanation{-9, 4}

The two diagonal values give {-9, 4}. Equal values are listed once; this is a set of distinct eigenvalues, not an algebraic-multiplicity count.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r04-eigenvalue-0029/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-eigenvalue-diagonal-set · candidate





## 30. Distinct Real Eigenvalues of a Diagonal Matrix

Over R, the 2×2 matrix is diag(-8,5). Give its distinct real eigenvalues as an increasing set with each value once.

View answer and explanation{-8, 5}

The two diagonal values give {-8, 5}. Equal values are listed once; this is a set of distinct eigenvalues, not an algebraic-multiplicity count.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r04-eigenvalue-0030/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-eigenvalue-diagonal-set · candidate





## 31. Distinct Real Eigenvalues of a Diagonal Matrix

Over R, the 2×2 matrix is diag(-7,6). Give its distinct real eigenvalues as an increasing set with each value once.

View answer and explanation{-7, 6}

The two diagonal values give {-7, 6}. Equal values are listed once; this is a set of distinct eigenvalues, not an algebraic-multiplicity count.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r04-eigenvalue-0031/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-eigenvalue-diagonal-set · candidate





## 32. Distinct Real Eigenvalues of a Diagonal Matrix

Over R, the 2×2 matrix is diag(-6,7). Give its distinct real eigenvalues as an increasing set with each value once.

View answer and explanation{-6, 7}

The two diagonal values give {-6, 7}. Equal values are listed once; this is a set of distinct eigenvalues, not an algebraic-multiplicity count.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r04-eigenvalue-0032/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-eigenvalue-diagonal-set · candidate





## 33. Distinct Real Eigenvalues of a Diagonal Matrix

Over R, the 2×2 matrix is diag(-5,8). Give its distinct real eigenvalues as an increasing set with each value once.

View answer and explanation{-5, 8}

The two diagonal values give {-5, 8}. Equal values are listed once; this is a set of distinct eigenvalues, not an algebraic-multiplicity count.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r04-eigenvalue-0033/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-eigenvalue-diagonal-set · candidate





## 34. Distinct Real Eigenvalues of a Diagonal Matrix

Over R, the 2×2 matrix is diag(-4,9). Give its distinct real eigenvalues as an increasing set with each value once.

View answer and explanation{-4, 9}

The two diagonal values give {-4, 9}. Equal values are listed once; this is a set of distinct eigenvalues, not an algebraic-multiplicity count.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r04-eigenvalue-0034/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-eigenvalue-diagonal-set · candidate





## 35. Distinct Real Eigenvalues of a Diagonal Matrix

Over R, the 2×2 matrix is diag(-3,10). Give its distinct real eigenvalues as an increasing set with each value once.

View answer and explanation{-3, 10}

The two diagonal values give {-3, 10}. Equal values are listed once; this is a set of distinct eigenvalues, not an algebraic-multiplicity count.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r04-eigenvalue-0035/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-eigenvalue-diagonal-set · candidate





## 36. Distinct Real Eigenvalues of a Diagonal Matrix

Over R, the 2×2 matrix is diag(-2,11). Give its distinct real eigenvalues as an increasing set with each value once.

View answer and explanation{-2, 11}

The two diagonal values give {-2, 11}. Equal values are listed once; this is a set of distinct eigenvalues, not an algebraic-multiplicity count.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r04-eigenvalue-0036/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-eigenvalue-diagonal-set · candidate





## 37. Distinct Real Eigenvalues of a Diagonal Matrix

Over R, the 2×2 matrix is diag(-1,12). Give its distinct real eigenvalues as an increasing set with each value once.

View answer and explanation{-1, 12}

The two diagonal values give {-1, 12}. Equal values are listed once; this is a set of distinct eigenvalues, not an algebraic-multiplicity count.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r04-eigenvalue-0037/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-eigenvalue-diagonal-set · candidate





## 38. Distinct Real Eigenvalues of a Diagonal Matrix

Over R, the 2×2 matrix is diag(0,13). Give its distinct real eigenvalues as an increasing set with each value once.

View answer and explanation{0, 13}

The two diagonal values give {0, 13}. Equal values are listed once; this is a set of distinct eigenvalues, not an algebraic-multiplicity count.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r04-eigenvalue-0038/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-eigenvalue-diagonal-set · candidate





## 39. Distinct Real Eigenvalues of a Diagonal Matrix

Over R, the 2×2 matrix is diag(1,14). Give its distinct real eigenvalues as an increasing set with each value once.

View answer and explanation{1, 14}

The two diagonal values give {1, 14}. Equal values are listed once; this is a set of distinct eigenvalues, not an algebraic-multiplicity count.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r04-eigenvalue-0039/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-eigenvalue-diagonal-set · candidate





## 40. Distinct Real Eigenvalues of a Diagonal Matrix

Over R, the 2×2 matrix is diag(2,15). Give its distinct real eigenvalues as an increasing set with each value once.

View answer and explanation{2, 15}

The two diagonal values give {2, 15}. Equal values are listed once; this is a set of distinct eigenvalues, not an algebraic-multiplicity count.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r04-eigenvalue-0040/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-eigenvalue-diagonal-set · candidate





## 41. Distinct Real Eigenvalues of a Diagonal Matrix

Over R, the 2×2 matrix is diag(3,16). Give its distinct real eigenvalues as an increasing set with each value once.

View answer and explanation{3, 16}

The two diagonal values give {3, 16}. Equal values are listed once; this is a set of distinct eigenvalues, not an algebraic-multiplicity count.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r04-eigenvalue-0041/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-eigenvalue-diagonal-set · candidate





## 42. Distinct Real Eigenvalues of a Diagonal Matrix

Over R, the 2×2 matrix is diag(4,17). Give its distinct real eigenvalues as an increasing set with each value once.

View answer and explanation{4, 17}

The two diagonal values give {4, 17}. Equal values are listed once; this is a set of distinct eigenvalues, not an algebraic-multiplicity count.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r04-eigenvalue-0042/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-eigenvalue-diagonal-set · candidate





## 43. Distinct Real Eigenvalues of a Diagonal Matrix

Over R, the 2×2 matrix is diag(5,18). Give its distinct real eigenvalues as an increasing set with each value once.

View answer and explanation{5, 18}

The two diagonal values give {5, 18}. Equal values are listed once; this is a set of distinct eigenvalues, not an algebraic-multiplicity count.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r04-eigenvalue-0043/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-eigenvalue-diagonal-set · candidate





## 44. Distinct Real Eigenvalues of a Diagonal Matrix

Over R, the 2×2 matrix is diag(6,19). Give its distinct real eigenvalues as an increasing set with each value once.

View answer and explanation{6, 19}

The two diagonal values give {6, 19}. Equal values are listed once; this is a set of distinct eigenvalues, not an algebraic-multiplicity count.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r04-eigenvalue-0044/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-eigenvalue-diagonal-set · candidate





## 45. Distinct Real Eigenvalues of a Diagonal Matrix

Over R, the 2×2 matrix is diag(7,20). Give its distinct real eigenvalues as an increasing set with each value once.

View answer and explanation{7, 20}

The two diagonal values give {7, 20}. Equal values are listed once; this is a set of distinct eigenvalues, not an algebraic-multiplicity count.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r04-eigenvalue-0045/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-eigenvalue-diagonal-set · candidate





## 46. Distinct Real Eigenvalues of a Diagonal Matrix

Over R, the 2×2 matrix is diag(8,21). Give its distinct real eigenvalues as an increasing set with each value once.

View answer and explanation{8, 21}

The two diagonal values give {8, 21}. Equal values are listed once; this is a set of distinct eigenvalues, not an algebraic-multiplicity count.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r04-eigenvalue-0046/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-eigenvalue-diagonal-set · candidate





## 47. Distinct Real Eigenvalues of a Diagonal Matrix

Over R, the 2×2 matrix is diag(9,22). Give its distinct real eigenvalues as an increasing set with each value once.

View answer and explanation{9, 22}

The two diagonal values give {9, 22}. Equal values are listed once; this is a set of distinct eigenvalues, not an algebraic-multiplicity count.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r04-eigenvalue-0047/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-eigenvalue-diagonal-set · candidate





## 48. Distinct Real Eigenvalues of a Diagonal Matrix

Over R, the 2×2 matrix is diag(10,23). Give its distinct real eigenvalues as an increasing set with each value once.

View answer and explanation{10, 23}

The two diagonal values give {10, 23}. Equal values are listed once; this is a set of distinct eigenvalues, not an algebraic-multiplicity count.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r04-eigenvalue-0048/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-eigenvalue-diagonal-set · candidate





## 49. Distinct Real Eigenvalues of a Diagonal Matrix

Over R, the 2×2 matrix is diag(11,24). Give its distinct real eigenvalues as an increasing set with each value once.

View answer and explanation{11, 24}

The two diagonal values give {11, 24}. Equal values are listed once; this is a set of distinct eigenvalues, not an algebraic-multiplicity count.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r04-eigenvalue-0049/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-eigenvalue-diagonal-set · candidate





## 50. Distinct Real Eigenvalues of a Diagonal Matrix

Over R, the 2×2 matrix is diag(12,25). Give its distinct real eigenvalues as an increasing set with each value once.

View answer and explanation{12, 25}

The two diagonal values give {12, 25}. Equal values are listed once; this is a set of distinct eigenvalues, not an algebraic-multiplicity count.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r04-eigenvalue-0050/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-eigenvalue-diagonal-set · candidate





## 51. Kernel Membership for a Candidate Scalar

Over R, the 2×2 matrix is M=[[2, 1], [0, 3]]. For the candidate scalar λ=99 and v=[0, 0], does v belong to ker(M−λI)? Do not assume λ is an eigenvalue.

View answer and explanationYes

The residual (M−λI)v is [0, 0], giving Yes. Every such kernel contains zero, and zero-vector membership does not establish that λ is an eigenvalue.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r04-eigenvalue-0051/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-eigenvalue-kernel-membership · candidate





## 52. Kernel Membership for a Candidate Scalar

Over R, the 2×2 matrix is M=[[-11, 0], [0, 3]]. For the candidate scalar λ=3 and v=[0, 1], does v belong to ker(M−λI)? Do not assume λ is an eigenvalue.

View answer and explanationYes

The residual (M−λI)v is [0, 0], giving Yes. Every such kernel contains zero, and zero-vector membership does not establish that λ is an eigenvalue.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r04-eigenvalue-0052/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-eigenvalue-kernel-membership · candidate





## 53. Kernel Membership for a Candidate Scalar

Over R, the 2×2 matrix is M=[[-10, 0], [0, 4]]. For the candidate scalar λ=-10 and v=[1, 1], does v belong to ker(M−λI)? Do not assume λ is an eigenvalue.

View answer and explanationNo

The residual (M−λI)v is [0, 14], giving No. Every such kernel contains zero, and zero-vector membership does not establish that λ is an eigenvalue.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r04-eigenvalue-0053/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-eigenvalue-kernel-membership · candidate





## 54. Kernel Membership for a Candidate Scalar

Over R, the 2×2 matrix is M=[[-9, 0], [0, 5]]. For the candidate scalar λ=-8 and v=[1, -1], does v belong to ker(M−λI)? Do not assume λ is an eigenvalue.

View answer and explanationNo

The residual (M−λI)v is [-1, -13], giving No. Every such kernel contains zero, and zero-vector membership does not establish that λ is an eigenvalue.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r04-eigenvalue-0054/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-eigenvalue-kernel-membership · candidate





## 55. Kernel Membership for a Candidate Scalar

Over R, the 2×2 matrix is M=[[4, 1], [1, 4]]. For the candidate scalar λ=5 and v=[1, 1], does v belong to ker(M−λI)? Do not assume λ is an eigenvalue.

View answer and explanationYes

The residual (M−λI)v is [0, 0], giving Yes. Every such kernel contains zero, and zero-vector membership does not establish that λ is an eigenvalue.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r04-eigenvalue-0055/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-eigenvalue-kernel-membership · candidate





## 56. Kernel Membership for a Candidate Scalar

Over R, the 2×2 matrix is M=[[5, 0], [1, 6]]. For the candidate scalar λ=5 and v=[1, -1], does v belong to ker(M−λI)? Do not assume λ is an eigenvalue.

View answer and explanationYes

The residual (M−λI)v is [0, 0], giving Yes. Every such kernel contains zero, and zero-vector membership does not establish that λ is an eigenvalue.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r04-eigenvalue-0056/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-eigenvalue-kernel-membership · candidate





## 57. Kernel Membership for a Candidate Scalar

Over R, the 2×2 matrix is M=[[-6, 0], [0, 8]]. For the candidate scalar λ=-6 and v=[1, 1], does v belong to ker(M−λI)? Do not assume λ is an eigenvalue.

View answer and explanationNo

The residual (M−λI)v is [0, 14], giving No. Every such kernel contains zero, and zero-vector membership does not establish that λ is an eigenvalue.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r04-eigenvalue-0057/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-eigenvalue-kernel-membership · candidate





## 58. Kernel Membership for a Candidate Scalar

Over R, the 2×2 matrix is M=[[-5, 0], [0, 9]]. For the candidate scalar λ=-4 and v=[1, -1], does v belong to ker(M−λI)? Do not assume λ is an eigenvalue.

View answer and explanationNo

The residual (M−λI)v is [-1, -13], giving No. Every such kernel contains zero, and zero-vector membership does not establish that λ is an eigenvalue.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r04-eigenvalue-0058/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-eigenvalue-kernel-membership · candidate





## 59. Kernel Membership for a Candidate Scalar

Over R, the 2×2 matrix is M=[[-4, 0], [0, 10]]. For the candidate scalar λ=-4 and v=[1, 0], does v belong to ker(M−λI)? Do not assume λ is an eigenvalue.

View answer and explanationYes

The residual (M−λI)v is [0, 0], giving Yes. Every such kernel contains zero, and zero-vector membership does not establish that λ is an eigenvalue.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r04-eigenvalue-0059/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-eigenvalue-kernel-membership · candidate





## 60. Kernel Membership for a Candidate Scalar

Over R, the 2×2 matrix is M=[[-3, 0], [0, 11]]. For the candidate scalar λ=11 and v=[0, 1], does v belong to ker(M−λI)? Do not assume λ is an eigenvalue.

View answer and explanationYes

The residual (M−λI)v is [0, 0], giving Yes. Every such kernel contains zero, and zero-vector membership does not establish that λ is an eigenvalue.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r04-eigenvalue-0060/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-eigenvalue-kernel-membership · candidate





## 61. Kernel Membership for a Candidate Scalar

Over R, the 2×2 matrix is M=[[-2, 0], [0, 12]]. For the candidate scalar λ=-2 and v=[1, 1], does v belong to ker(M−λI)? Do not assume λ is an eigenvalue.

View answer and explanationNo

The residual (M−λI)v is [0, 14], giving No. Every such kernel contains zero, and zero-vector membership does not establish that λ is an eigenvalue.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r04-eigenvalue-0061/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-eigenvalue-kernel-membership · candidate





## 62. Kernel Membership for a Candidate Scalar

Over R, the 2×2 matrix is M=[[-1, 0], [0, 13]]. For the candidate scalar λ=0 and v=[1, -1], does v belong to ker(M−λI)? Do not assume λ is an eigenvalue.

View answer and explanationNo

The residual (M−λI)v is [-1, -13], giving No. Every such kernel contains zero, and zero-vector membership does not establish that λ is an eigenvalue.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r04-eigenvalue-0062/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-eigenvalue-kernel-membership · candidate





## 63. Kernel Membership for a Candidate Scalar

Over R, the 2×2 matrix is M=[[0, 0], [0, 14]]. For the candidate scalar λ=0 and v=[1, 0], does v belong to ker(M−λI)? Do not assume λ is an eigenvalue.

View answer and explanationYes

The residual (M−λI)v is [0, 0], giving Yes. Every such kernel contains zero, and zero-vector membership does not establish that λ is an eigenvalue.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r04-eigenvalue-0063/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-eigenvalue-kernel-membership · candidate





## 64. Kernel Membership for a Candidate Scalar

Over R, the 2×2 matrix is M=[[1, 0], [0, 15]]. For the candidate scalar λ=15 and v=[0, 1], does v belong to ker(M−λI)? Do not assume λ is an eigenvalue.

View answer and explanationYes

The residual (M−λI)v is [0, 0], giving Yes. Every such kernel contains zero, and zero-vector membership does not establish that λ is an eigenvalue.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r04-eigenvalue-0064/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-eigenvalue-kernel-membership · candidate





## 65. Kernel Membership for a Candidate Scalar

Over R, the 2×2 matrix is M=[[2, 0], [0, 16]]. For the candidate scalar λ=2 and v=[1, 1], does v belong to ker(M−λI)? Do not assume λ is an eigenvalue.

View answer and explanationNo

The residual (M−λI)v is [0, 14], giving No. Every such kernel contains zero, and zero-vector membership does not establish that λ is an eigenvalue.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r04-eigenvalue-0065/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-eigenvalue-kernel-membership · candidate





## 66. Kernel Membership for a Candidate Scalar

Over R, the 2×2 matrix is M=[[3, 0], [0, 17]]. For the candidate scalar λ=4 and v=[1, -1], does v belong to ker(M−λI)? Do not assume λ is an eigenvalue.

View answer and explanationNo

The residual (M−λI)v is [-1, -13], giving No. Every such kernel contains zero, and zero-vector membership does not establish that λ is an eigenvalue.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r04-eigenvalue-0066/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-eigenvalue-kernel-membership · candidate





## 67. Kernel Membership for a Candidate Scalar

Over R, the 2×2 matrix is M=[[4, 0], [0, 18]]. For the candidate scalar λ=4 and v=[1, 0], does v belong to ker(M−λI)? Do not assume λ is an eigenvalue.

View answer and explanationYes

The residual (M−λI)v is [0, 0], giving Yes. Every such kernel contains zero, and zero-vector membership does not establish that λ is an eigenvalue.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r04-eigenvalue-0067/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-eigenvalue-kernel-membership · candidate





## 68. Kernel Membership for a Candidate Scalar

Over R, the 2×2 matrix is M=[[5, 0], [0, 19]]. For the candidate scalar λ=19 and v=[0, 1], does v belong to ker(M−λI)? Do not assume λ is an eigenvalue.

View answer and explanationYes

The residual (M−λI)v is [0, 0], giving Yes. Every such kernel contains zero, and zero-vector membership does not establish that λ is an eigenvalue.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r04-eigenvalue-0068/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-eigenvalue-kernel-membership · candidate





## 69. Kernel Membership for a Candidate Scalar

Over R, the 2×2 matrix is M=[[6, 0], [0, 20]]. For the candidate scalar λ=6 and v=[1, 1], does v belong to ker(M−λI)? Do not assume λ is an eigenvalue.

View answer and explanationNo

The residual (M−λI)v is [0, 14], giving No. Every such kernel contains zero, and zero-vector membership does not establish that λ is an eigenvalue.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r04-eigenvalue-0069/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-eigenvalue-kernel-membership · candidate





## 70. Kernel Membership for a Candidate Scalar

Over R, the 2×2 matrix is M=[[7, 0], [0, 21]]. For the candidate scalar λ=8 and v=[1, -1], does v belong to ker(M−λI)? Do not assume λ is an eigenvalue.

View answer and explanationNo

The residual (M−λI)v is [-1, -13], giving No. Every such kernel contains zero, and zero-vector membership does not establish that λ is an eigenvalue.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r04-eigenvalue-0070/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-eigenvalue-kernel-membership · candidate





## 71. Kernel Membership for a Candidate Scalar

Over R, the 2×2 matrix is M=[[8, 0], [0, 22]]. For the candidate scalar λ=8 and v=[1, 0], does v belong to ker(M−λI)? Do not assume λ is an eigenvalue.

View answer and explanationYes

The residual (M−λI)v is [0, 0], giving Yes. Every such kernel contains zero, and zero-vector membership does not establish that λ is an eigenvalue.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r04-eigenvalue-0071/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-eigenvalue-kernel-membership · candidate





## 72. Kernel Membership for a Candidate Scalar

Over R, the 2×2 matrix is M=[[9, 0], [0, 23]]. For the candidate scalar λ=23 and v=[0, 1], does v belong to ker(M−λI)? Do not assume λ is an eigenvalue.

View answer and explanationYes

The residual (M−λI)v is [0, 0], giving Yes. Every such kernel contains zero, and zero-vector membership does not establish that λ is an eigenvalue.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r04-eigenvalue-0072/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-eigenvalue-kernel-membership · candidate





## 73. Kernel Membership for a Candidate Scalar

Over R, the 2×2 matrix is M=[[10, 0], [0, 24]]. For the candidate scalar λ=10 and v=[1, 1], does v belong to ker(M−λI)? Do not assume λ is an eigenvalue.

View answer and explanationNo

The residual (M−λI)v is [0, 14], giving No. Every such kernel contains zero, and zero-vector membership does not establish that λ is an eigenvalue.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r04-eigenvalue-0073/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-eigenvalue-kernel-membership · candidate





## 74. Kernel Membership for a Candidate Scalar

Over R, the 2×2 matrix is M=[[11, 0], [0, 25]]. For the candidate scalar λ=12 and v=[1, -1], does v belong to ker(M−λI)? Do not assume λ is an eigenvalue.

View answer and explanationNo

The residual (M−λI)v is [-1, -13], giving No. Every such kernel contains zero, and zero-vector membership does not establish that λ is an eigenvalue.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r04-eigenvalue-0074/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-eigenvalue-kernel-membership · candidate





## 75. Kernel Membership for a Candidate Scalar

Over R, the 2×2 matrix is M=[[12, 0], [0, 26]]. For the candidate scalar λ=12 and v=[1, 0], does v belong to ker(M−λI)? Do not assume λ is an eigenvalue.

View answer and explanationYes

The residual (M−λI)v is [0, 0], giving Yes. Every such kernel contains zero, and zero-vector membership does not establish that λ is an eigenvalue.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r04-eigenvalue-0075/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-eigenvalue-kernel-membership · candidate





## 76. Number of Distinct Real Eigenvalues of a 2×2 Matrix

Over R, the 2×2 matrix is M=[[0, 0], [0, 0]]. Find only the number of distinct real eigenvalues: zero, one, or two. Do not output approximate roots or count algebraic multiplicity.

View answer and explanation1

The characteristic polynomial coefficients are (1,0,0), with discriminant Δ=0. Negative, zero, and positive Δ give zero, one, and two distinct real roots, so the answer is 1. A positive nonsquare Δ still gives two real roots; no diagonalizability, complex-spectrum, or Jordan conclusion follows here.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r04-eigenvalue-0076/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-eigenvalue-real-count · candidate





## 77. Number of Distinct Real Eigenvalues of a 2×2 Matrix

Over R, the 2×2 matrix is M=[[1, -1], [2, 1]]. Find only the number of distinct real eigenvalues: zero, one, or two. Do not output approximate roots or count algebraic multiplicity.

View answer and explanation0

The characteristic polynomial coefficients are (1,-2,3), with discriminant Δ=-8. Negative, zero, and positive Δ give zero, one, and two distinct real roots, so the answer is 0. A positive nonsquare Δ still gives two real roots; no diagonalizability, complex-spectrum, or Jordan conclusion follows here.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r04-eigenvalue-0077/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-eigenvalue-real-count · candidate





## 78. Number of Distinct Real Eigenvalues of a 2×2 Matrix

Over R, the 2×2 matrix is M=[[1, 1], [0, 1]]. Find only the number of distinct real eigenvalues: zero, one, or two. Do not output approximate roots or count algebraic multiplicity.

View answer and explanation1

The characteristic polynomial coefficients are (1,-2,1), with discriminant Δ=0. Negative, zero, and positive Δ give zero, one, and two distinct real roots, so the answer is 1. A positive nonsquare Δ still gives two real roots; no diagonalizability, complex-spectrum, or Jordan conclusion follows here.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r04-eigenvalue-0078/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-eigenvalue-real-count · candidate





## 79. Number of Distinct Real Eigenvalues of a 2×2 Matrix

Over R, the 2×2 matrix is M=[[-1, 1], [1, 1]]. Find only the number of distinct real eigenvalues: zero, one, or two. Do not output approximate roots or count algebraic multiplicity.

View answer and explanation2

The characteristic polynomial coefficients are (1,0,-2), with discriminant Δ=8. Negative, zero, and positive Δ give zero, one, and two distinct real roots, so the answer is 2. A positive nonsquare Δ still gives two real roots; no diagonalizability, complex-spectrum, or Jordan conclusion follows here.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r04-eigenvalue-0079/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-eigenvalue-real-count · candidate





## 80. Number of Distinct Real Eigenvalues of a 2×2 Matrix

Over R, the 2×2 matrix is M=[[2, -1], [3, 2]]. Find only the number of distinct real eigenvalues: zero, one, or two. Do not output approximate roots or count algebraic multiplicity.

View answer and explanation0

The characteristic polynomial coefficients are (1,-4,7), with discriminant Δ=-12. Negative, zero, and positive Δ give zero, one, and two distinct real roots, so the answer is 0. A positive nonsquare Δ still gives two real roots; no diagonalizability, complex-spectrum, or Jordan conclusion follows here.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r04-eigenvalue-0080/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-eigenvalue-real-count · candidate





## 81. Number of Distinct Real Eigenvalues of a 2×2 Matrix

Over R, the 2×2 matrix is M=[[2, 1], [0, 2]]. Find only the number of distinct real eigenvalues: zero, one, or two. Do not output approximate roots or count algebraic multiplicity.

View answer and explanation1

The characteristic polynomial coefficients are (1,-4,4), with discriminant Δ=0. Negative, zero, and positive Δ give zero, one, and two distinct real roots, so the answer is 1. A positive nonsquare Δ still gives two real roots; no diagonalizability, complex-spectrum, or Jordan conclusion follows here.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r04-eigenvalue-0081/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-eigenvalue-real-count · candidate





## 82. Number of Distinct Real Eigenvalues of a 2×2 Matrix

Over R, the 2×2 matrix is M=[[-2, 1], [1, 2]]. Find only the number of distinct real eigenvalues: zero, one, or two. Do not output approximate roots or count algebraic multiplicity.

View answer and explanation2

The characteristic polynomial coefficients are (1,0,-5), with discriminant Δ=20. Negative, zero, and positive Δ give zero, one, and two distinct real roots, so the answer is 2. A positive nonsquare Δ still gives two real roots; no diagonalizability, complex-spectrum, or Jordan conclusion follows here.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r04-eigenvalue-0082/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-eigenvalue-real-count · candidate





## 83. Number of Distinct Real Eigenvalues of a 2×2 Matrix

Over R, the 2×2 matrix is M=[[3, -1], [4, 3]]. Find only the number of distinct real eigenvalues: zero, one, or two. Do not output approximate roots or count algebraic multiplicity.

View answer and explanation0

The characteristic polynomial coefficients are (1,-6,13), with discriminant Δ=-16. Negative, zero, and positive Δ give zero, one, and two distinct real roots, so the answer is 0. A positive nonsquare Δ still gives two real roots; no diagonalizability, complex-spectrum, or Jordan conclusion follows here.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r04-eigenvalue-0083/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-eigenvalue-real-count · candidate





## 84. Number of Distinct Real Eigenvalues of a 2×2 Matrix

Over R, the 2×2 matrix is M=[[3, 1], [0, 3]]. Find only the number of distinct real eigenvalues: zero, one, or two. Do not output approximate roots or count algebraic multiplicity.

View answer and explanation1

The characteristic polynomial coefficients are (1,-6,9), with discriminant Δ=0. Negative, zero, and positive Δ give zero, one, and two distinct real roots, so the answer is 1. A positive nonsquare Δ still gives two real roots; no diagonalizability, complex-spectrum, or Jordan conclusion follows here.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r04-eigenvalue-0084/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-eigenvalue-real-count · candidate





## 85. Number of Distinct Real Eigenvalues of a 2×2 Matrix

Over R, the 2×2 matrix is M=[[-3, 1], [1, 3]]. Find only the number of distinct real eigenvalues: zero, one, or two. Do not output approximate roots or count algebraic multiplicity.

View answer and explanation2

The characteristic polynomial coefficients are (1,0,-10), with discriminant Δ=40. Negative, zero, and positive Δ give zero, one, and two distinct real roots, so the answer is 2. A positive nonsquare Δ still gives two real roots; no diagonalizability, complex-spectrum, or Jordan conclusion follows here.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r04-eigenvalue-0085/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-eigenvalue-real-count · candidate





## 86. Number of Distinct Real Eigenvalues of a 2×2 Matrix

Over R, the 2×2 matrix is M=[[4, -1], [5, 4]]. Find only the number of distinct real eigenvalues: zero, one, or two. Do not output approximate roots or count algebraic multiplicity.

View answer and explanation0

The characteristic polynomial coefficients are (1,-8,21), with discriminant Δ=-20. Negative, zero, and positive Δ give zero, one, and two distinct real roots, so the answer is 0. A positive nonsquare Δ still gives two real roots; no diagonalizability, complex-spectrum, or Jordan conclusion follows here.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r04-eigenvalue-0086/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-eigenvalue-real-count · candidate





## 87. Number of Distinct Real Eigenvalues of a 2×2 Matrix

Over R, the 2×2 matrix is M=[[4, 1], [0, 4]]. Find only the number of distinct real eigenvalues: zero, one, or two. Do not output approximate roots or count algebraic multiplicity.

View answer and explanation1

The characteristic polynomial coefficients are (1,-8,16), with discriminant Δ=0. Negative, zero, and positive Δ give zero, one, and two distinct real roots, so the answer is 1. A positive nonsquare Δ still gives two real roots; no diagonalizability, complex-spectrum, or Jordan conclusion follows here.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r04-eigenvalue-0087/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-eigenvalue-real-count · candidate





## 88. Number of Distinct Real Eigenvalues of a 2×2 Matrix

Over R, the 2×2 matrix is M=[[-4, 1], [1, 4]]. Find only the number of distinct real eigenvalues: zero, one, or two. Do not output approximate roots or count algebraic multiplicity.

View answer and explanation2

The characteristic polynomial coefficients are (1,0,-17), with discriminant Δ=68. Negative, zero, and positive Δ give zero, one, and two distinct real roots, so the answer is 2. A positive nonsquare Δ still gives two real roots; no diagonalizability, complex-spectrum, or Jordan conclusion follows here.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r04-eigenvalue-0088/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-eigenvalue-real-count · candidate





## 89. Number of Distinct Real Eigenvalues of a 2×2 Matrix

Over R, the 2×2 matrix is M=[[5, -1], [6, 5]]. Find only the number of distinct real eigenvalues: zero, one, or two. Do not output approximate roots or count algebraic multiplicity.

View answer and explanation0

The characteristic polynomial coefficients are (1,-10,31), with discriminant Δ=-24. Negative, zero, and positive Δ give zero, one, and two distinct real roots, so the answer is 0. A positive nonsquare Δ still gives two real roots; no diagonalizability, complex-spectrum, or Jordan conclusion follows here.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r04-eigenvalue-0089/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-eigenvalue-real-count · candidate





## 90. Number of Distinct Real Eigenvalues of a 2×2 Matrix

Over R, the 2×2 matrix is M=[[5, 1], [0, 5]]. Find only the number of distinct real eigenvalues: zero, one, or two. Do not output approximate roots or count algebraic multiplicity.

View answer and explanation1

The characteristic polynomial coefficients are (1,-10,25), with discriminant Δ=0. Negative, zero, and positive Δ give zero, one, and two distinct real roots, so the answer is 1. A positive nonsquare Δ still gives two real roots; no diagonalizability, complex-spectrum, or Jordan conclusion follows here.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r04-eigenvalue-0090/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-eigenvalue-real-count · candidate





## 91. Number of Distinct Real Eigenvalues of a 2×2 Matrix

Over R, the 2×2 matrix is M=[[-5, 1], [1, 5]]. Find only the number of distinct real eigenvalues: zero, one, or two. Do not output approximate roots or count algebraic multiplicity.

View answer and explanation2

The characteristic polynomial coefficients are (1,0,-26), with discriminant Δ=104. Negative, zero, and positive Δ give zero, one, and two distinct real roots, so the answer is 2. A positive nonsquare Δ still gives two real roots; no diagonalizability, complex-spectrum, or Jordan conclusion follows here.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r04-eigenvalue-0091/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-eigenvalue-real-count · candidate





## 92. Number of Distinct Real Eigenvalues of a 2×2 Matrix

Over R, the 2×2 matrix is M=[[6, -1], [7, 6]]. Find only the number of distinct real eigenvalues: zero, one, or two. Do not output approximate roots or count algebraic multiplicity.

View answer and explanation0

The characteristic polynomial coefficients are (1,-12,43), with discriminant Δ=-28. Negative, zero, and positive Δ give zero, one, and two distinct real roots, so the answer is 0. A positive nonsquare Δ still gives two real roots; no diagonalizability, complex-spectrum, or Jordan conclusion follows here.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r04-eigenvalue-0092/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-eigenvalue-real-count · candidate





## 93. Number of Distinct Real Eigenvalues of a 2×2 Matrix

Over R, the 2×2 matrix is M=[[6, 1], [0, 6]]. Find only the number of distinct real eigenvalues: zero, one, or two. Do not output approximate roots or count algebraic multiplicity.

View answer and explanation1

The characteristic polynomial coefficients are (1,-12,36), with discriminant Δ=0. Negative, zero, and positive Δ give zero, one, and two distinct real roots, so the answer is 1. A positive nonsquare Δ still gives two real roots; no diagonalizability, complex-spectrum, or Jordan conclusion follows here.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r04-eigenvalue-0093/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-eigenvalue-real-count · candidate





## 94. Number of Distinct Real Eigenvalues of a 2×2 Matrix

Over R, the 2×2 matrix is M=[[-6, 1], [1, 6]]. Find only the number of distinct real eigenvalues: zero, one, or two. Do not output approximate roots or count algebraic multiplicity.

View answer and explanation2

The characteristic polynomial coefficients are (1,0,-37), with discriminant Δ=148. Negative, zero, and positive Δ give zero, one, and two distinct real roots, so the answer is 2. A positive nonsquare Δ still gives two real roots; no diagonalizability, complex-spectrum, or Jordan conclusion follows here.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r04-eigenvalue-0094/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-eigenvalue-real-count · candidate





## 95. Number of Distinct Real Eigenvalues of a 2×2 Matrix

Over R, the 2×2 matrix is M=[[7, -1], [8, 7]]. Find only the number of distinct real eigenvalues: zero, one, or two. Do not output approximate roots or count algebraic multiplicity.

View answer and explanation0

The characteristic polynomial coefficients are (1,-14,57), with discriminant Δ=-32. Negative, zero, and positive Δ give zero, one, and two distinct real roots, so the answer is 0. A positive nonsquare Δ still gives two real roots; no diagonalizability, complex-spectrum, or Jordan conclusion follows here.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r04-eigenvalue-0095/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-eigenvalue-real-count · candidate





## 96. Number of Distinct Real Eigenvalues of a 2×2 Matrix

Over R, the 2×2 matrix is M=[[7, 1], [0, 7]]. Find only the number of distinct real eigenvalues: zero, one, or two. Do not output approximate roots or count algebraic multiplicity.

View answer and explanation1

The characteristic polynomial coefficients are (1,-14,49), with discriminant Δ=0. Negative, zero, and positive Δ give zero, one, and two distinct real roots, so the answer is 1. A positive nonsquare Δ still gives two real roots; no diagonalizability, complex-spectrum, or Jordan conclusion follows here.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r04-eigenvalue-0096/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-eigenvalue-real-count · candidate





## 97. Number of Distinct Real Eigenvalues of a 2×2 Matrix

Over R, the 2×2 matrix is M=[[-7, 1], [1, 7]]. Find only the number of distinct real eigenvalues: zero, one, or two. Do not output approximate roots or count algebraic multiplicity.

View answer and explanation2

The characteristic polynomial coefficients are (1,0,-50), with discriminant Δ=200. Negative, zero, and positive Δ give zero, one, and two distinct real roots, so the answer is 2. A positive nonsquare Δ still gives two real roots; no diagonalizability, complex-spectrum, or Jordan conclusion follows here.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r04-eigenvalue-0097/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-eigenvalue-real-count · candidate





## 98. Number of Distinct Real Eigenvalues of a 2×2 Matrix

Over R, the 2×2 matrix is M=[[8, -1], [9, 8]]. Find only the number of distinct real eigenvalues: zero, one, or two. Do not output approximate roots or count algebraic multiplicity.

View answer and explanation0

The characteristic polynomial coefficients are (1,-16,73), with discriminant Δ=-36. Negative, zero, and positive Δ give zero, one, and two distinct real roots, so the answer is 0. A positive nonsquare Δ still gives two real roots; no diagonalizability, complex-spectrum, or Jordan conclusion follows here.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r04-eigenvalue-0098/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-eigenvalue-real-count · candidate





## 99. Number of Distinct Real Eigenvalues of a 2×2 Matrix

Over R, the 2×2 matrix is M=[[8, 1], [0, 8]]. Find only the number of distinct real eigenvalues: zero, one, or two. Do not output approximate roots or count algebraic multiplicity.

View answer and explanation1

The characteristic polynomial coefficients are (1,-16,64), with discriminant Δ=0. Negative, zero, and positive Δ give zero, one, and two distinct real roots, so the answer is 1. A positive nonsquare Δ still gives two real roots; no diagonalizability, complex-spectrum, or Jordan conclusion follows here.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r04-eigenvalue-0099/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-eigenvalue-real-count · candidate





## 100. Number of Distinct Real Eigenvalues of a 2×2 Matrix

Over R, the 2×2 matrix is M=[[-8, 1], [1, 8]]. Find only the number of distinct real eigenvalues: zero, one, or two. Do not output approximate roots or count algebraic multiplicity.

View answer and explanation2

The characteristic polynomial coefficients are (1,0,-65), with discriminant Δ=260. Negative, zero, and positive Δ give zero, one, and two distinct real roots, so the answer is 2. A positive nonsquare Δ still gives two real roots; no diagonalizability, complex-spectrum, or Jordan conclusion follows here.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r04-eigenvalue-0100/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-eigenvalue-real-count · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261006-advance-20261007-r04-eigenvalue-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261006-advance-20261007-r04-eigenvalue-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261006-advance-20261007-r04-eigenvalue-foundations/validation-report.json)