# Real 2×2 Characteristic Polynomial and Residual Depth

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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-20261007-advance-20261009-r07-eigenvalue-foundations · 2026-10-07 · 100 questions

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



## 1. Descending Characteristic Coefficients of a Real 2×2 Matrix

Over R, M=[[0, 0], [0, 0]] is 2×2. Give its characteristic polynomial coefficients in descending order (λ²,λ,constant).

View answer and explanation(1, 0, 0)

Use (1,−tr(M),det(M)) to obtain (1, 0, 0), not the ascending order used for the separate polynomial lists. No diagonalization or multiplicity claim is added.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r07-eigenvalue-0001/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-density-eigen-characteristic-coefficients · candidate





## 2. Descending Characteristic Coefficients of a Real 2×2 Matrix

Over R, M=[[1, 0], [0, 1]] is 2×2. Give its characteristic polynomial coefficients in descending order (λ²,λ,constant).

View answer and explanation(1, -2, 1)

Use (1,−tr(M),det(M)) to obtain (1, -2, 1), not the ascending order used for the separate polynomial lists. No diagonalization or multiplicity claim is added.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r07-eigenvalue-0002/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-density-eigen-characteristic-coefficients · candidate





## 3. Descending Characteristic Coefficients of a Real 2×2 Matrix

Over R, M=[[-10, 3], [1, 3]] is 2×2. Give its characteristic polynomial coefficients in descending order (λ²,λ,constant).

View answer and explanation(1, 7, -33)

Use (1,−tr(M),det(M)) to obtain (1, 7, -33), not the ascending order used for the separate polynomial lists. No diagonalization or multiplicity claim is added.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r07-eigenvalue-0003/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-density-eigen-characteristic-coefficients · candidate





## 4. Descending Characteristic Coefficients of a Real 2×2 Matrix

Over R, M=[[-9, 1], [-1, 4]] is 2×2. Give its characteristic polynomial coefficients in descending order (λ²,λ,constant).

View answer and explanation(1, 5, -35)

Use (1,−tr(M),det(M)) to obtain (1, 5, -35), not the ascending order used for the separate polynomial lists. No diagonalization or multiplicity claim is added.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r07-eigenvalue-0004/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-density-eigen-characteristic-coefficients · candidate





## 5. Descending Characteristic Coefficients of a Real 2×2 Matrix

Over R, M=[[-8, 2], [1, 5]] is 2×2. Give its characteristic polynomial coefficients in descending order (λ²,λ,constant).

View answer and explanation(1, 3, -42)

Use (1,−tr(M),det(M)) to obtain (1, 3, -42), not the ascending order used for the separate polynomial lists. No diagonalization or multiplicity claim is added.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r07-eigenvalue-0005/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-density-eigen-characteristic-coefficients · candidate





## 6. Descending Characteristic Coefficients of a Real 2×2 Matrix

Over R, M=[[-7, 3], [-1, 6]] is 2×2. Give its characteristic polynomial coefficients in descending order (λ²,λ,constant).

View answer and explanation(1, 1, -39)

Use (1,−tr(M),det(M)) to obtain (1, 1, -39), not the ascending order used for the separate polynomial lists. No diagonalization or multiplicity claim is added.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r07-eigenvalue-0006/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-density-eigen-characteristic-coefficients · candidate





## 7. Descending Characteristic Coefficients of a Real 2×2 Matrix

Over R, M=[[-6, 1], [1, 7]] is 2×2. Give its characteristic polynomial coefficients in descending order (λ²,λ,constant).

View answer and explanation(1, -1, -43)

Use (1,−tr(M),det(M)) to obtain (1, -1, -43), not the ascending order used for the separate polynomial lists. No diagonalization or multiplicity claim is added.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r07-eigenvalue-0007/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-density-eigen-characteristic-coefficients · candidate





## 8. Descending Characteristic Coefficients of a Real 2×2 Matrix

Over R, M=[[-5, 2], [-1, 8]] is 2×2. Give its characteristic polynomial coefficients in descending order (λ²,λ,constant).

View answer and explanation(1, -3, -38)

Use (1,−tr(M),det(M)) to obtain (1, -3, -38), not the ascending order used for the separate polynomial lists. No diagonalization or multiplicity claim is added.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r07-eigenvalue-0008/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-density-eigen-characteristic-coefficients · candidate





## 9. Descending Characteristic Coefficients of a Real 2×2 Matrix

Over R, M=[[-4, 3], [1, 9]] is 2×2. Give its characteristic polynomial coefficients in descending order (λ²,λ,constant).

View answer and explanation(1, -5, -39)

Use (1,−tr(M),det(M)) to obtain (1, -5, -39), not the ascending order used for the separate polynomial lists. No diagonalization or multiplicity claim is added.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r07-eigenvalue-0009/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-density-eigen-characteristic-coefficients · candidate





## 10. Descending Characteristic Coefficients of a Real 2×2 Matrix

Over R, M=[[-3, 1], [-1, 10]] is 2×2. Give its characteristic polynomial coefficients in descending order (λ²,λ,constant).

View answer and explanation(1, -7, -29)

Use (1,−tr(M),det(M)) to obtain (1, -7, -29), not the ascending order used for the separate polynomial lists. No diagonalization or multiplicity claim is added.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r07-eigenvalue-0010/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-density-eigen-characteristic-coefficients · candidate





## 11. Descending Characteristic Coefficients of a Real 2×2 Matrix

Over R, M=[[-2, 2], [1, 11]] is 2×2. Give its characteristic polynomial coefficients in descending order (λ²,λ,constant).

View answer and explanation(1, -9, -24)

Use (1,−tr(M),det(M)) to obtain (1, -9, -24), not the ascending order used for the separate polynomial lists. No diagonalization or multiplicity claim is added.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r07-eigenvalue-0011/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-density-eigen-characteristic-coefficients · candidate





## 12. Descending Characteristic Coefficients of a Real 2×2 Matrix

Over R, M=[[-1, 3], [-1, 12]] is 2×2. Give its characteristic polynomial coefficients in descending order (λ²,λ,constant).

View answer and explanation(1, -11, -9)

Use (1,−tr(M),det(M)) to obtain (1, -11, -9), not the ascending order used for the separate polynomial lists. No diagonalization or multiplicity claim is added.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r07-eigenvalue-0012/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-density-eigen-characteristic-coefficients · candidate





## 13. Descending Characteristic Coefficients of a Real 2×2 Matrix

Over R, M=[[0, 1], [1, 13]] is 2×2. Give its characteristic polynomial coefficients in descending order (λ²,λ,constant).

View answer and explanation(1, -13, -1)

Use (1,−tr(M),det(M)) to obtain (1, -13, -1), not the ascending order used for the separate polynomial lists. No diagonalization or multiplicity claim is added.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r07-eigenvalue-0013/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-density-eigen-characteristic-coefficients · candidate





## 14. Descending Characteristic Coefficients of a Real 2×2 Matrix

Over R, M=[[1, 2], [-1, 14]] is 2×2. Give its characteristic polynomial coefficients in descending order (λ²,λ,constant).

View answer and explanation(1, -15, 16)

Use (1,−tr(M),det(M)) to obtain (1, -15, 16), not the ascending order used for the separate polynomial lists. No diagonalization or multiplicity claim is added.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r07-eigenvalue-0014/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-density-eigen-characteristic-coefficients · candidate





## 15. Descending Characteristic Coefficients of a Real 2×2 Matrix

Over R, M=[[2, 3], [1, 15]] is 2×2. Give its characteristic polynomial coefficients in descending order (λ²,λ,constant).

View answer and explanation(1, -17, 27)

Use (1,−tr(M),det(M)) to obtain (1, -17, 27), not the ascending order used for the separate polynomial lists. No diagonalization or multiplicity claim is added.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r07-eigenvalue-0015/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-density-eigen-characteristic-coefficients · candidate





## 16. Descending Characteristic Coefficients of a Real 2×2 Matrix

Over R, M=[[3, 1], [-1, 16]] is 2×2. Give its characteristic polynomial coefficients in descending order (λ²,λ,constant).

View answer and explanation(1, -19, 49)

Use (1,−tr(M),det(M)) to obtain (1, -19, 49), not the ascending order used for the separate polynomial lists. No diagonalization or multiplicity claim is added.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r07-eigenvalue-0016/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-density-eigen-characteristic-coefficients · candidate





## 17. Descending Characteristic Coefficients of a Real 2×2 Matrix

Over R, M=[[4, 2], [1, 17]] is 2×2. Give its characteristic polynomial coefficients in descending order (λ²,λ,constant).

View answer and explanation(1, -21, 66)

Use (1,−tr(M),det(M)) to obtain (1, -21, 66), not the ascending order used for the separate polynomial lists. No diagonalization or multiplicity claim is added.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r07-eigenvalue-0017/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-density-eigen-characteristic-coefficients · candidate





## 18. Descending Characteristic Coefficients of a Real 2×2 Matrix

Over R, M=[[5, 3], [-1, 18]] is 2×2. Give its characteristic polynomial coefficients in descending order (λ²,λ,constant).

View answer and explanation(1, -23, 93)

Use (1,−tr(M),det(M)) to obtain (1, -23, 93), not the ascending order used for the separate polynomial lists. No diagonalization or multiplicity claim is added.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r07-eigenvalue-0018/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-density-eigen-characteristic-coefficients · candidate





## 19. Descending Characteristic Coefficients of a Real 2×2 Matrix

Over R, M=[[6, 1], [1, 19]] is 2×2. Give its characteristic polynomial coefficients in descending order (λ²,λ,constant).

View answer and explanation(1, -25, 113)

Use (1,−tr(M),det(M)) to obtain (1, -25, 113), not the ascending order used for the separate polynomial lists. No diagonalization or multiplicity claim is added.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r07-eigenvalue-0019/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-density-eigen-characteristic-coefficients · candidate





## 20. Descending Characteristic Coefficients of a Real 2×2 Matrix

Over R, M=[[7, 2], [-1, 20]] is 2×2. Give its characteristic polynomial coefficients in descending order (λ²,λ,constant).

View answer and explanation(1, -27, 142)

Use (1,−tr(M),det(M)) to obtain (1, -27, 142), not the ascending order used for the separate polynomial lists. No diagonalization or multiplicity claim is added.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r07-eigenvalue-0020/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-density-eigen-characteristic-coefficients · candidate





## 21. Descending Characteristic Coefficients of a Real 2×2 Matrix

Over R, M=[[8, 3], [1, 21]] is 2×2. Give its characteristic polynomial coefficients in descending order (λ²,λ,constant).

View answer and explanation(1, -29, 165)

Use (1,−tr(M),det(M)) to obtain (1, -29, 165), not the ascending order used for the separate polynomial lists. No diagonalization or multiplicity claim is added.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r07-eigenvalue-0021/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-density-eigen-characteristic-coefficients · candidate





## 22. Descending Characteristic Coefficients of a Real 2×2 Matrix

Over R, M=[[9, 1], [-1, 22]] is 2×2. Give its characteristic polynomial coefficients in descending order (λ²,λ,constant).

View answer and explanation(1, -31, 199)

Use (1,−tr(M),det(M)) to obtain (1, -31, 199), not the ascending order used for the separate polynomial lists. No diagonalization or multiplicity claim is added.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r07-eigenvalue-0022/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-density-eigen-characteristic-coefficients · candidate





## 23. Descending Characteristic Coefficients of a Real 2×2 Matrix

Over R, M=[[10, 2], [1, 23]] is 2×2. Give its characteristic polynomial coefficients in descending order (λ²,λ,constant).

View answer and explanation(1, -33, 228)

Use (1,−tr(M),det(M)) to obtain (1, -33, 228), not the ascending order used for the separate polynomial lists. No diagonalization or multiplicity claim is added.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r07-eigenvalue-0023/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-density-eigen-characteristic-coefficients · candidate





## 24. Descending Characteristic Coefficients of a Real 2×2 Matrix

Over R, M=[[11, 3], [-1, 24]] is 2×2. Give its characteristic polynomial coefficients in descending order (λ²,λ,constant).

View answer and explanation(1, -35, 267)

Use (1,−tr(M),det(M)) to obtain (1, -35, 267), not the ascending order used for the separate polynomial lists. No diagonalization or multiplicity claim is added.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r07-eigenvalue-0024/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-density-eigen-characteristic-coefficients · candidate





## 25. Descending Characteristic Coefficients of a Real 2×2 Matrix

Over R, M=[[12, 1], [1, 25]] is 2×2. Give its characteristic polynomial coefficients in descending order (λ²,λ,constant).

View answer and explanation(1, -37, 299)

Use (1,−tr(M),det(M)) to obtain (1, -37, 299), not the ascending order used for the separate polynomial lists. No diagonalization or multiplicity claim is added.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r07-eigenvalue-0025/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-density-eigen-characteristic-coefficients · candidate





## 26. Exact Real 2×2 Characteristic Discriminant

Over R, M=[[0, -1], [1, 0]] is 2×2. Compute the exact characteristic discriminant tr(M)²−4det(M).

View answer and explanation-4

The integer discriminant is -4. Its sign controls the number of distinct real eigenvalues; no floating roots or complex-spectrum expansion are asserted.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r07-eigenvalue-0026/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-density-eigen-discriminant · candidate





## 27. Exact Real 2×2 Characteristic Discriminant

Over R, M=[[3, 1], [0, 3]] is 2×2. Compute the exact characteristic discriminant tr(M)²−4det(M).

View answer and explanation0

The integer discriminant is 0. Its sign controls the number of distinct real eigenvalues; no floating roots or complex-spectrum expansion are asserted.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r07-eigenvalue-0027/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-density-eigen-discriminant · candidate





## 28. Exact Real 2×2 Characteristic Discriminant

Over R, M=[[2, 1], [2, 0]] is 2×2. Compute the exact characteristic discriminant tr(M)²−4det(M).

View answer and explanation12

The integer discriminant is 12. Its sign controls the number of distinct real eigenvalues; no floating roots or complex-spectrum expansion are asserted.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r07-eigenvalue-0028/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-density-eigen-discriminant · candidate





## 29. Exact Real 2×2 Characteristic Discriminant

Over R, M=[[3, 1], [-3, 0]] is 2×2. Compute the exact characteristic discriminant tr(M)²−4det(M).

View answer and explanation-3

The integer discriminant is -3. Its sign controls the number of distinct real eigenvalues; no floating roots or complex-spectrum expansion are asserted.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r07-eigenvalue-0029/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-density-eigen-discriminant · candidate





## 30. Exact Real 2×2 Characteristic Discriminant

Over R, M=[[4, 1], [4, 0]] is 2×2. Compute the exact characteristic discriminant tr(M)²−4det(M).

View answer and explanation32

The integer discriminant is 32. Its sign controls the number of distinct real eigenvalues; no floating roots or complex-spectrum expansion are asserted.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r07-eigenvalue-0030/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-density-eigen-discriminant · candidate





## 31. Exact Real 2×2 Characteristic Discriminant

Over R, M=[[5, 1], [-5, 0]] is 2×2. Compute the exact characteristic discriminant tr(M)²−4det(M).

View answer and explanation5

The integer discriminant is 5. Its sign controls the number of distinct real eigenvalues; no floating roots or complex-spectrum expansion are asserted.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r07-eigenvalue-0031/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-density-eigen-discriminant · candidate





## 32. Exact Real 2×2 Characteristic Discriminant

Over R, M=[[6, 1], [6, 0]] is 2×2. Compute the exact characteristic discriminant tr(M)²−4det(M).

View answer and explanation60

The integer discriminant is 60. Its sign controls the number of distinct real eigenvalues; no floating roots or complex-spectrum expansion are asserted.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r07-eigenvalue-0032/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-density-eigen-discriminant · candidate





## 33. Exact Real 2×2 Characteristic Discriminant

Over R, M=[[7, 1], [-7, 0]] is 2×2. Compute the exact characteristic discriminant tr(M)²−4det(M).

View answer and explanation21

The integer discriminant is 21. Its sign controls the number of distinct real eigenvalues; no floating roots or complex-spectrum expansion are asserted.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r07-eigenvalue-0033/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-density-eigen-discriminant · candidate





## 34. Exact Real 2×2 Characteristic Discriminant

Over R, M=[[8, 1], [8, 0]] is 2×2. Compute the exact characteristic discriminant tr(M)²−4det(M).

View answer and explanation96

The integer discriminant is 96. Its sign controls the number of distinct real eigenvalues; no floating roots or complex-spectrum expansion are asserted.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r07-eigenvalue-0034/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-density-eigen-discriminant · candidate





## 35. Exact Real 2×2 Characteristic Discriminant

Over R, M=[[9, 1], [-9, 0]] is 2×2. Compute the exact characteristic discriminant tr(M)²−4det(M).

View answer and explanation45

The integer discriminant is 45. Its sign controls the number of distinct real eigenvalues; no floating roots or complex-spectrum expansion are asserted.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r07-eigenvalue-0035/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-density-eigen-discriminant · candidate





## 36. Exact Real 2×2 Characteristic Discriminant

Over R, M=[[10, 1], [10, 0]] is 2×2. Compute the exact characteristic discriminant tr(M)²−4det(M).

View answer and explanation140

The integer discriminant is 140. Its sign controls the number of distinct real eigenvalues; no floating roots or complex-spectrum expansion are asserted.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r07-eigenvalue-0036/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-density-eigen-discriminant · candidate





## 37. Exact Real 2×2 Characteristic Discriminant

Over R, M=[[11, 1], [-11, 0]] is 2×2. Compute the exact characteristic discriminant tr(M)²−4det(M).

View answer and explanation77

The integer discriminant is 77. Its sign controls the number of distinct real eigenvalues; no floating roots or complex-spectrum expansion are asserted.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r07-eigenvalue-0037/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-density-eigen-discriminant · candidate





## 38. Exact Real 2×2 Characteristic Discriminant

Over R, M=[[12, 1], [12, 0]] is 2×2. Compute the exact characteristic discriminant tr(M)²−4det(M).

View answer and explanation192

The integer discriminant is 192. Its sign controls the number of distinct real eigenvalues; no floating roots or complex-spectrum expansion are asserted.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r07-eigenvalue-0038/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-density-eigen-discriminant · candidate





## 39. Exact Real 2×2 Characteristic Discriminant

Over R, M=[[13, 1], [-13, 0]] is 2×2. Compute the exact characteristic discriminant tr(M)²−4det(M).

View answer and explanation117

The integer discriminant is 117. Its sign controls the number of distinct real eigenvalues; no floating roots or complex-spectrum expansion are asserted.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r07-eigenvalue-0039/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-density-eigen-discriminant · candidate





## 40. Exact Real 2×2 Characteristic Discriminant

Over R, M=[[14, 1], [14, 0]] is 2×2. Compute the exact characteristic discriminant tr(M)²−4det(M).

View answer and explanation252

The integer discriminant is 252. Its sign controls the number of distinct real eigenvalues; no floating roots or complex-spectrum expansion are asserted.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r07-eigenvalue-0040/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-density-eigen-discriminant · candidate





## 41. Exact Real 2×2 Characteristic Discriminant

Over R, M=[[15, 1], [-15, 0]] is 2×2. Compute the exact characteristic discriminant tr(M)²−4det(M).

View answer and explanation165

The integer discriminant is 165. Its sign controls the number of distinct real eigenvalues; no floating roots or complex-spectrum expansion are asserted.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r07-eigenvalue-0041/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-density-eigen-discriminant · candidate





## 42. Exact Real 2×2 Characteristic Discriminant

Over R, M=[[16, 1], [16, 0]] is 2×2. Compute the exact characteristic discriminant tr(M)²−4det(M).

View answer and explanation320

The integer discriminant is 320. Its sign controls the number of distinct real eigenvalues; no floating roots or complex-spectrum expansion are asserted.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r07-eigenvalue-0042/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-density-eigen-discriminant · candidate





## 43. Exact Real 2×2 Characteristic Discriminant

Over R, M=[[17, 1], [-17, 0]] is 2×2. Compute the exact characteristic discriminant tr(M)²−4det(M).

View answer and explanation221

The integer discriminant is 221. Its sign controls the number of distinct real eigenvalues; no floating roots or complex-spectrum expansion are asserted.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r07-eigenvalue-0043/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-density-eigen-discriminant · candidate





## 44. Exact Real 2×2 Characteristic Discriminant

Over R, M=[[18, 1], [18, 0]] is 2×2. Compute the exact characteristic discriminant tr(M)²−4det(M).

View answer and explanation396

The integer discriminant is 396. Its sign controls the number of distinct real eigenvalues; no floating roots or complex-spectrum expansion are asserted.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r07-eigenvalue-0044/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-density-eigen-discriminant · candidate





## 45. Exact Real 2×2 Characteristic Discriminant

Over R, M=[[19, 1], [-19, 0]] is 2×2. Compute the exact characteristic discriminant tr(M)²−4det(M).

View answer and explanation285

The integer discriminant is 285. Its sign controls the number of distinct real eigenvalues; no floating roots or complex-spectrum expansion are asserted.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r07-eigenvalue-0045/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-density-eigen-discriminant · candidate





## 46. Exact Real 2×2 Characteristic Discriminant

Over R, M=[[20, 1], [20, 0]] is 2×2. Compute the exact characteristic discriminant tr(M)²−4det(M).

View answer and explanation480

The integer discriminant is 480. Its sign controls the number of distinct real eigenvalues; no floating roots or complex-spectrum expansion are asserted.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r07-eigenvalue-0046/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-density-eigen-discriminant · candidate





## 47. Exact Real 2×2 Characteristic Discriminant

Over R, M=[[21, 1], [-21, 0]] is 2×2. Compute the exact characteristic discriminant tr(M)²−4det(M).

View answer and explanation357

The integer discriminant is 357. Its sign controls the number of distinct real eigenvalues; no floating roots or complex-spectrum expansion are asserted.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r07-eigenvalue-0047/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-density-eigen-discriminant · candidate





## 48. Exact Real 2×2 Characteristic Discriminant

Over R, M=[[22, 1], [22, 0]] is 2×2. Compute the exact characteristic discriminant tr(M)²−4det(M).

View answer and explanation572

The integer discriminant is 572. Its sign controls the number of distinct real eigenvalues; no floating roots or complex-spectrum expansion are asserted.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r07-eigenvalue-0048/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-density-eigen-discriminant · candidate





## 49. Exact Real 2×2 Characteristic Discriminant

Over R, M=[[23, 1], [-23, 0]] is 2×2. Compute the exact characteristic discriminant tr(M)²−4det(M).

View answer and explanation437

The integer discriminant is 437. Its sign controls the number of distinct real eigenvalues; no floating roots or complex-spectrum expansion are asserted.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r07-eigenvalue-0049/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-density-eigen-discriminant · candidate





## 50. Exact Real 2×2 Characteristic Discriminant

Over R, M=[[24, 1], [24, 0]] is 2×2. Compute the exact characteristic discriminant tr(M)²−4det(M).

View answer and explanation672

The integer discriminant is 672. Its sign controls the number of distinct real eigenvalues; no floating roots or complex-spectrum expansion are asserted.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r07-eigenvalue-0050/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-density-eigen-discriminant · candidate





## 51. Ordered Residual of a Specified Kernel Equation

Over R, M=[[2, 1], [0, -3]] is 2×2, λ=4, and v=[0, 0]. Give the ordered residual Mv−λv, without assuming λ is an eigenvalue.

View answer and explanation(0, 0)

Exact multiplication and subtraction yield (0, 0). A zero vector always has zero residual, which alone does not establish an eigenvalue or an eligible eigenvector.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r07-eigenvalue-0051/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-density-eigen-kernel-residual · candidate





## 52. Ordered Residual of a Specified Kernel Equation

Over R, M=[[2, 0], [0, 3]] is 2×2, λ=2, and v=[1, 0]. Give the ordered residual Mv−λv, without assuming λ is an eigenvalue.

View answer and explanation(0, 0)

Exact multiplication and subtraction yield (0, 0). A zero vector always has zero residual, which alone does not establish an eigenvalue or an eligible eigenvector.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r07-eigenvalue-0052/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-density-eigen-kernel-residual · candidate





## 53. Ordered Residual of a Specified Kernel Equation

Over R, M=[[3, 3], [1, 0]] is 2×2, λ=0, and v=[3, 0]. Give the ordered residual Mv−λv, without assuming λ is an eigenvalue.

View answer and explanation(9, 3)

Exact multiplication and subtraction yield (9, 3). A zero vector always has zero residual, which alone does not establish an eigenvalue or an eligible eigenvector.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r07-eigenvalue-0053/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-density-eigen-kernel-residual · candidate





## 54. Ordered Residual of a Specified Kernel Equation

Over R, M=[[4, 1], [-1, -1]] is 2×2, λ=1, and v=[1, -1]. Give the ordered residual Mv−λv, without assuming λ is an eigenvalue.

View answer and explanation(2, 1)

Exact multiplication and subtraction yield (2, 1). A zero vector always has zero residual, which alone does not establish an eigenvalue or an eligible eigenvector.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r07-eigenvalue-0054/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-density-eigen-kernel-residual · candidate





## 55. Ordered Residual of a Specified Kernel Equation

Over R, M=[[5, 2], [1, -2]] is 2×2, λ=2, and v=[2, -2]. Give the ordered residual Mv−λv, without assuming λ is an eigenvalue.

View answer and explanation(2, 10)

Exact multiplication and subtraction yield (2, 10). A zero vector always has zero residual, which alone does not establish an eigenvalue or an eligible eigenvector.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r07-eigenvalue-0055/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-density-eigen-kernel-residual · candidate





## 56. Ordered Residual of a Specified Kernel Equation

Over R, M=[[6, 3], [-1, -3]] is 2×2, λ=-2, and v=[3, 2]. Give the ordered residual Mv−λv, without assuming λ is an eigenvalue.

View answer and explanation(30, -5)

Exact multiplication and subtraction yield (30, -5). A zero vector always has zero residual, which alone does not establish an eigenvalue or an eligible eigenvector.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r07-eigenvalue-0056/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-density-eigen-kernel-residual · candidate





## 57. Ordered Residual of a Specified Kernel Equation

Over R, M=[[7, 1], [1, -4]] is 2×2, λ=-1, and v=[1, 1]. Give the ordered residual Mv−λv, without assuming λ is an eigenvalue.

View answer and explanation(9, -2)

Exact multiplication and subtraction yield (9, -2). A zero vector always has zero residual, which alone does not establish an eigenvalue or an eligible eigenvector.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r07-eigenvalue-0057/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-density-eigen-kernel-residual · candidate





## 58. Ordered Residual of a Specified Kernel Equation

Over R, M=[[8, 2], [-1, -5]] is 2×2, λ=0, and v=[2, 0]. Give the ordered residual Mv−λv, without assuming λ is an eigenvalue.

View answer and explanation(16, -2)

Exact multiplication and subtraction yield (16, -2). A zero vector always has zero residual, which alone does not establish an eigenvalue or an eligible eigenvector.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r07-eigenvalue-0058/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-density-eigen-kernel-residual · candidate





## 59. Ordered Residual of a Specified Kernel Equation

Over R, M=[[9, 3], [1, -6]] is 2×2, λ=1, and v=[3, -1]. Give the ordered residual Mv−λv, without assuming λ is an eigenvalue.

View answer and explanation(21, 10)

Exact multiplication and subtraction yield (21, 10). A zero vector always has zero residual, which alone does not establish an eigenvalue or an eligible eigenvector.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r07-eigenvalue-0059/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-density-eigen-kernel-residual · candidate





## 60. Ordered Residual of a Specified Kernel Equation

Over R, M=[[10, 1], [-1, -7]] is 2×2, λ=2, and v=[1, -2]. Give the ordered residual Mv−λv, without assuming λ is an eigenvalue.

View answer and explanation(6, 17)

Exact multiplication and subtraction yield (6, 17). A zero vector always has zero residual, which alone does not establish an eigenvalue or an eligible eigenvector.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r07-eigenvalue-0060/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-density-eigen-kernel-residual · candidate





## 61. Ordered Residual of a Specified Kernel Equation

Over R, M=[[11, 2], [1, -8]] is 2×2, λ=-2, and v=[2, 2]. Give the ordered residual Mv−λv, without assuming λ is an eigenvalue.

View answer and explanation(30, -10)

Exact multiplication and subtraction yield (30, -10). A zero vector always has zero residual, which alone does not establish an eigenvalue or an eligible eigenvector.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r07-eigenvalue-0061/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-density-eigen-kernel-residual · candidate





## 62. Ordered Residual of a Specified Kernel Equation

Over R, M=[[12, 3], [-1, -9]] is 2×2, λ=-1, and v=[3, 1]. Give the ordered residual Mv−λv, without assuming λ is an eigenvalue.

View answer and explanation(42, -11)

Exact multiplication and subtraction yield (42, -11). A zero vector always has zero residual, which alone does not establish an eigenvalue or an eligible eigenvector.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r07-eigenvalue-0062/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-density-eigen-kernel-residual · candidate





## 63. Ordered Residual of a Specified Kernel Equation

Over R, M=[[13, 1], [1, -10]] is 2×2, λ=0, and v=[1, 0]. Give the ordered residual Mv−λv, without assuming λ is an eigenvalue.

View answer and explanation(13, 1)

Exact multiplication and subtraction yield (13, 1). A zero vector always has zero residual, which alone does not establish an eigenvalue or an eligible eigenvector.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r07-eigenvalue-0063/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-density-eigen-kernel-residual · candidate





## 64. Ordered Residual of a Specified Kernel Equation

Over R, M=[[14, 2], [-1, -11]] is 2×2, λ=1, and v=[2, -1]. Give the ordered residual Mv−λv, without assuming λ is an eigenvalue.

View answer and explanation(24, 10)

Exact multiplication and subtraction yield (24, 10). A zero vector always has zero residual, which alone does not establish an eigenvalue or an eligible eigenvector.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r07-eigenvalue-0064/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-density-eigen-kernel-residual · candidate





## 65. Ordered Residual of a Specified Kernel Equation

Over R, M=[[15, 3], [1, -12]] is 2×2, λ=2, and v=[3, -2]. Give the ordered residual Mv−λv, without assuming λ is an eigenvalue.

View answer and explanation(33, 31)

Exact multiplication and subtraction yield (33, 31). A zero vector always has zero residual, which alone does not establish an eigenvalue or an eligible eigenvector.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r07-eigenvalue-0065/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-density-eigen-kernel-residual · candidate





## 66. Ordered Residual of a Specified Kernel Equation

Over R, M=[[16, 1], [-1, -13]] is 2×2, λ=-2, and v=[1, 2]. Give the ordered residual Mv−λv, without assuming λ is an eigenvalue.

View answer and explanation(20, -23)

Exact multiplication and subtraction yield (20, -23). A zero vector always has zero residual, which alone does not establish an eigenvalue or an eligible eigenvector.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r07-eigenvalue-0066/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-density-eigen-kernel-residual · candidate





## 67. Ordered Residual of a Specified Kernel Equation

Over R, M=[[17, 2], [1, -14]] is 2×2, λ=-1, and v=[2, 1]. Give the ordered residual Mv−λv, without assuming λ is an eigenvalue.

View answer and explanation(38, -11)

Exact multiplication and subtraction yield (38, -11). A zero vector always has zero residual, which alone does not establish an eigenvalue or an eligible eigenvector.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r07-eigenvalue-0067/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-density-eigen-kernel-residual · candidate





## 68. Ordered Residual of a Specified Kernel Equation

Over R, M=[[18, 3], [-1, -15]] is 2×2, λ=0, and v=[3, 0]. Give the ordered residual Mv−λv, without assuming λ is an eigenvalue.

View answer and explanation(54, -3)

Exact multiplication and subtraction yield (54, -3). A zero vector always has zero residual, which alone does not establish an eigenvalue or an eligible eigenvector.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r07-eigenvalue-0068/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-density-eigen-kernel-residual · candidate





## 69. Ordered Residual of a Specified Kernel Equation

Over R, M=[[19, 1], [1, -16]] is 2×2, λ=1, and v=[1, -1]. Give the ordered residual Mv−λv, without assuming λ is an eigenvalue.

View answer and explanation(17, 18)

Exact multiplication and subtraction yield (17, 18). A zero vector always has zero residual, which alone does not establish an eigenvalue or an eligible eigenvector.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r07-eigenvalue-0069/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-density-eigen-kernel-residual · candidate





## 70. Ordered Residual of a Specified Kernel Equation

Over R, M=[[20, 2], [-1, -17]] is 2×2, λ=2, and v=[2, -2]. Give the ordered residual Mv−λv, without assuming λ is an eigenvalue.

View answer and explanation(32, 36)

Exact multiplication and subtraction yield (32, 36). A zero vector always has zero residual, which alone does not establish an eigenvalue or an eligible eigenvector.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r07-eigenvalue-0070/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-density-eigen-kernel-residual · candidate





## 71. Ordered Residual of a Specified Kernel Equation

Over R, M=[[21, 3], [1, -18]] is 2×2, λ=-2, and v=[3, 2]. Give the ordered residual Mv−λv, without assuming λ is an eigenvalue.

View answer and explanation(75, -29)

Exact multiplication and subtraction yield (75, -29). A zero vector always has zero residual, which alone does not establish an eigenvalue or an eligible eigenvector.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r07-eigenvalue-0071/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-density-eigen-kernel-residual · candidate





## 72. Ordered Residual of a Specified Kernel Equation

Over R, M=[[22, 1], [-1, -19]] is 2×2, λ=-1, and v=[1, 1]. Give the ordered residual Mv−λv, without assuming λ is an eigenvalue.

View answer and explanation(24, -19)

Exact multiplication and subtraction yield (24, -19). A zero vector always has zero residual, which alone does not establish an eigenvalue or an eligible eigenvector.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r07-eigenvalue-0072/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-density-eigen-kernel-residual · candidate





## 73. Ordered Residual of a Specified Kernel Equation

Over R, M=[[23, 2], [1, -20]] is 2×2, λ=0, and v=[2, 0]. Give the ordered residual Mv−λv, without assuming λ is an eigenvalue.

View answer and explanation(46, 2)

Exact multiplication and subtraction yield (46, 2). A zero vector always has zero residual, which alone does not establish an eigenvalue or an eligible eigenvector.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r07-eigenvalue-0073/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-density-eigen-kernel-residual · candidate





## 74. Ordered Residual of a Specified Kernel Equation

Over R, M=[[24, 3], [-1, -21]] is 2×2, λ=1, and v=[3, -1]. Give the ordered residual Mv−λv, without assuming λ is an eigenvalue.

View answer and explanation(66, 19)

Exact multiplication and subtraction yield (66, 19). A zero vector always has zero residual, which alone does not establish an eigenvalue or an eligible eigenvector.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r07-eigenvalue-0074/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-density-eigen-kernel-residual · candidate





## 75. Ordered Residual of a Specified Kernel Equation

Over R, M=[[25, 1], [1, -22]] is 2×2, λ=2, and v=[1, -2]. Give the ordered residual Mv−λv, without assuming λ is an eigenvalue.

View answer and explanation(21, 49)

Exact multiplication and subtraction yield (21, 49). A zero vector always has zero residual, which alone does not establish an eigenvalue or an eligible eigenvector.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r07-eigenvalue-0075/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-density-eigen-kernel-residual · candidate





## 76. First Mismatch of a Nonzero Eigenvector Candidate

Over R, M=[[1, 1], [0, 2]] is 2×2, λ=2, and the nonzero candidate v=[1, 1]. Check residual components in order 1,2 and report the first nonzero component index, or none if both vanish.

View answer and explanationnone

The first mismatch is none. Only with the stated nonzero-vector premise does no mismatch support this eigenpair; this does not establish diagonalization or general spectral theory.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r07-eigenvalue-0076/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-density-eigen-first-mismatch · candidate





## 77. First Mismatch of a Nonzero Eigenvector Candidate

Over R, M=[[0, 2], [1, 2]] is 2×2, λ=0, and the nonzero candidate v=[0, 1]. Check residual components in order 1,2 and report the first nonzero component index, or none if both vanish.

View answer and explanation1

The first mismatch is 1. Only with the stated nonzero-vector premise does no mismatch support this eigenpair; this does not establish diagonalization or general spectral theory.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r07-eigenvalue-0077/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-density-eigen-first-mismatch · candidate





## 78. First Mismatch of a Nonzero Eigenvector Candidate

Over R, M=[[3, 0], [0, 8]] is 2×2, λ=3, and the nonzero candidate v=[1, 1]. Check residual components in order 1,2 and report the first nonzero component index, or none if both vanish.

View answer and explanation2

The first mismatch is 2. Only with the stated nonzero-vector premise does no mismatch support this eigenpair; this does not establish diagonalization or general spectral theory.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r07-eigenvalue-0078/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-density-eigen-first-mismatch · candidate





## 79. First Mismatch of a Nonzero Eigenvector Candidate

Over R, M=[[1, 1], [0, 5]] is 2×2, λ=5, and the nonzero candidate v=[1, 4]. Check residual components in order 1,2 and report the first nonzero component index, or none if both vanish.

View answer and explanationnone

The first mismatch is none. Only with the stated nonzero-vector premise does no mismatch support this eigenpair; this does not establish diagonalization or general spectral theory.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r07-eigenvalue-0079/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-density-eigen-first-mismatch · candidate





## 80. First Mismatch of a Nonzero Eigenvector Candidate

Over R, M=[[0, 5], [1, 2]] is 2×2, λ=0, and the nonzero candidate v=[0, 1]. Check residual components in order 1,2 and report the first nonzero component index, or none if both vanish.

View answer and explanation1

The first mismatch is 1. Only with the stated nonzero-vector premise does no mismatch support this eigenpair; this does not establish diagonalization or general spectral theory.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r07-eigenvalue-0080/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-density-eigen-first-mismatch · candidate





## 81. First Mismatch of a Nonzero Eigenvector Candidate

Over R, M=[[6, 0], [0, 14]] is 2×2, λ=6, and the nonzero candidate v=[1, 1]. Check residual components in order 1,2 and report the first nonzero component index, or none if both vanish.

View answer and explanation2

The first mismatch is 2. Only with the stated nonzero-vector premise does no mismatch support this eigenpair; this does not establish diagonalization or general spectral theory.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r07-eigenvalue-0081/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-density-eigen-first-mismatch · candidate





## 82. First Mismatch of a Nonzero Eigenvector Candidate

Over R, M=[[1, 1], [0, 8]] is 2×2, λ=8, and the nonzero candidate v=[1, 7]. Check residual components in order 1,2 and report the first nonzero component index, or none if both vanish.

View answer and explanationnone

The first mismatch is none. Only with the stated nonzero-vector premise does no mismatch support this eigenpair; this does not establish diagonalization or general spectral theory.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r07-eigenvalue-0082/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-density-eigen-first-mismatch · candidate





## 83. First Mismatch of a Nonzero Eigenvector Candidate

Over R, M=[[0, 8], [1, 2]] is 2×2, λ=0, and the nonzero candidate v=[0, 1]. Check residual components in order 1,2 and report the first nonzero component index, or none if both vanish.

View answer and explanation1

The first mismatch is 1. Only with the stated nonzero-vector premise does no mismatch support this eigenpair; this does not establish diagonalization or general spectral theory.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r07-eigenvalue-0083/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-density-eigen-first-mismatch · candidate





## 84. First Mismatch of a Nonzero Eigenvector Candidate

Over R, M=[[9, 0], [0, 20]] is 2×2, λ=9, and the nonzero candidate v=[1, 1]. Check residual components in order 1,2 and report the first nonzero component index, or none if both vanish.

View answer and explanation2

The first mismatch is 2. Only with the stated nonzero-vector premise does no mismatch support this eigenpair; this does not establish diagonalization or general spectral theory.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r07-eigenvalue-0084/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-density-eigen-first-mismatch · candidate





## 85. First Mismatch of a Nonzero Eigenvector Candidate

Over R, M=[[1, 1], [0, 11]] is 2×2, λ=11, and the nonzero candidate v=[1, 10]. Check residual components in order 1,2 and report the first nonzero component index, or none if both vanish.

View answer and explanationnone

The first mismatch is none. Only with the stated nonzero-vector premise does no mismatch support this eigenpair; this does not establish diagonalization or general spectral theory.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r07-eigenvalue-0085/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-density-eigen-first-mismatch · candidate





## 86. First Mismatch of a Nonzero Eigenvector Candidate

Over R, M=[[0, 11], [1, 2]] is 2×2, λ=0, and the nonzero candidate v=[0, 1]. Check residual components in order 1,2 and report the first nonzero component index, or none if both vanish.

View answer and explanation1

The first mismatch is 1. Only with the stated nonzero-vector premise does no mismatch support this eigenpair; this does not establish diagonalization or general spectral theory.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r07-eigenvalue-0086/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-density-eigen-first-mismatch · candidate





## 87. First Mismatch of a Nonzero Eigenvector Candidate

Over R, M=[[12, 0], [0, 26]] is 2×2, λ=12, and the nonzero candidate v=[1, 1]. Check residual components in order 1,2 and report the first nonzero component index, or none if both vanish.

View answer and explanation2

The first mismatch is 2. Only with the stated nonzero-vector premise does no mismatch support this eigenpair; this does not establish diagonalization or general spectral theory.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r07-eigenvalue-0087/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-density-eigen-first-mismatch · candidate





## 88. First Mismatch of a Nonzero Eigenvector Candidate

Over R, M=[[1, 1], [0, 14]] is 2×2, λ=14, and the nonzero candidate v=[1, 13]. Check residual components in order 1,2 and report the first nonzero component index, or none if both vanish.

View answer and explanationnone

The first mismatch is none. Only with the stated nonzero-vector premise does no mismatch support this eigenpair; this does not establish diagonalization or general spectral theory.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r07-eigenvalue-0088/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-density-eigen-first-mismatch · candidate





## 89. First Mismatch of a Nonzero Eigenvector Candidate

Over R, M=[[0, 14], [1, 2]] is 2×2, λ=0, and the nonzero candidate v=[0, 1]. Check residual components in order 1,2 and report the first nonzero component index, or none if both vanish.

View answer and explanation1

The first mismatch is 1. Only with the stated nonzero-vector premise does no mismatch support this eigenpair; this does not establish diagonalization or general spectral theory.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r07-eigenvalue-0089/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-density-eigen-first-mismatch · candidate





## 90. First Mismatch of a Nonzero Eigenvector Candidate

Over R, M=[[15, 0], [0, 32]] is 2×2, λ=15, and the nonzero candidate v=[1, 1]. Check residual components in order 1,2 and report the first nonzero component index, or none if both vanish.

View answer and explanation2

The first mismatch is 2. Only with the stated nonzero-vector premise does no mismatch support this eigenpair; this does not establish diagonalization or general spectral theory.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r07-eigenvalue-0090/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-density-eigen-first-mismatch · candidate





## 91. First Mismatch of a Nonzero Eigenvector Candidate

Over R, M=[[1, 1], [0, 17]] is 2×2, λ=17, and the nonzero candidate v=[1, 16]. Check residual components in order 1,2 and report the first nonzero component index, or none if both vanish.

View answer and explanationnone

The first mismatch is none. Only with the stated nonzero-vector premise does no mismatch support this eigenpair; this does not establish diagonalization or general spectral theory.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r07-eigenvalue-0091/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-density-eigen-first-mismatch · candidate





## 92. First Mismatch of a Nonzero Eigenvector Candidate

Over R, M=[[0, 17], [1, 2]] is 2×2, λ=0, and the nonzero candidate v=[0, 1]. Check residual components in order 1,2 and report the first nonzero component index, or none if both vanish.

View answer and explanation1

The first mismatch is 1. Only with the stated nonzero-vector premise does no mismatch support this eigenpair; this does not establish diagonalization or general spectral theory.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r07-eigenvalue-0092/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-density-eigen-first-mismatch · candidate





## 93. First Mismatch of a Nonzero Eigenvector Candidate

Over R, M=[[18, 0], [0, 38]] is 2×2, λ=18, and the nonzero candidate v=[1, 1]. Check residual components in order 1,2 and report the first nonzero component index, or none if both vanish.

View answer and explanation2

The first mismatch is 2. Only with the stated nonzero-vector premise does no mismatch support this eigenpair; this does not establish diagonalization or general spectral theory.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r07-eigenvalue-0093/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-density-eigen-first-mismatch · candidate





## 94. First Mismatch of a Nonzero Eigenvector Candidate

Over R, M=[[1, 1], [0, 20]] is 2×2, λ=20, and the nonzero candidate v=[1, 19]. Check residual components in order 1,2 and report the first nonzero component index, or none if both vanish.

View answer and explanationnone

The first mismatch is none. Only with the stated nonzero-vector premise does no mismatch support this eigenpair; this does not establish diagonalization or general spectral theory.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r07-eigenvalue-0094/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-density-eigen-first-mismatch · candidate





## 95. First Mismatch of a Nonzero Eigenvector Candidate

Over R, M=[[0, 20], [1, 2]] is 2×2, λ=0, and the nonzero candidate v=[0, 1]. Check residual components in order 1,2 and report the first nonzero component index, or none if both vanish.

View answer and explanation1

The first mismatch is 1. Only with the stated nonzero-vector premise does no mismatch support this eigenpair; this does not establish diagonalization or general spectral theory.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r07-eigenvalue-0095/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-density-eigen-first-mismatch · candidate





## 96. First Mismatch of a Nonzero Eigenvector Candidate

Over R, M=[[21, 0], [0, 44]] is 2×2, λ=21, and the nonzero candidate v=[1, 1]. Check residual components in order 1,2 and report the first nonzero component index, or none if both vanish.

View answer and explanation2

The first mismatch is 2. Only with the stated nonzero-vector premise does no mismatch support this eigenpair; this does not establish diagonalization or general spectral theory.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r07-eigenvalue-0096/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-density-eigen-first-mismatch · candidate





## 97. First Mismatch of a Nonzero Eigenvector Candidate

Over R, M=[[1, 1], [0, 23]] is 2×2, λ=23, and the nonzero candidate v=[1, 22]. Check residual components in order 1,2 and report the first nonzero component index, or none if both vanish.

View answer and explanationnone

The first mismatch is none. Only with the stated nonzero-vector premise does no mismatch support this eigenpair; this does not establish diagonalization or general spectral theory.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r07-eigenvalue-0097/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-density-eigen-first-mismatch · candidate





## 98. First Mismatch of a Nonzero Eigenvector Candidate

Over R, M=[[0, 23], [1, 2]] is 2×2, λ=0, and the nonzero candidate v=[0, 1]. Check residual components in order 1,2 and report the first nonzero component index, or none if both vanish.

View answer and explanation1

The first mismatch is 1. Only with the stated nonzero-vector premise does no mismatch support this eigenpair; this does not establish diagonalization or general spectral theory.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r07-eigenvalue-0098/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-density-eigen-first-mismatch · candidate





## 99. First Mismatch of a Nonzero Eigenvector Candidate

Over R, M=[[24, 0], [0, 50]] is 2×2, λ=24, and the nonzero candidate v=[1, 1]. Check residual components in order 1,2 and report the first nonzero component index, or none if both vanish.

View answer and explanation2

The first mismatch is 2. Only with the stated nonzero-vector premise does no mismatch support this eigenpair; this does not establish diagonalization or general spectral theory.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r07-eigenvalue-0099/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-density-eigen-first-mismatch · candidate





## 100. First Mismatch of a Nonzero Eigenvector Candidate

Over R, M=[[1, 1], [0, 26]] is 2×2, λ=26, and the nonzero candidate v=[1, 25]. Check residual components in order 1,2 and report the first nonzero component index, or none if both vanish.

View answer and explanationnone

The first mismatch is none. Only with the stated nonzero-vector premise does no mismatch support this eigenpair; this does not establish diagonalization or general spectral theory.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r07-eigenvalue-0100/)

Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-density-eigen-first-mismatch · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261007-advance-20261009-r07-eigenvalue-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261007-advance-20261009-r07-eigenvalue-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261007-advance-20261009-r07-eigenvalue-foundations/validation-report.json)