Real 2×2 Characteristic Polynomial and Residual Depth
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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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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 explanation
0
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
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 explanation
12
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
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
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 explanation
32
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
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 explanation
5
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
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 explanation
60
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
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 explanation
21
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
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 explanation
96
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
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 explanation
45
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
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 explanation
140
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
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 explanation
77
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
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 explanation
192
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
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 explanation
117
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
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 explanation
252
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
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 explanation
165
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
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 explanation
320
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
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 explanation
221
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
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 explanation
396
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
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 explanation
285
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
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 explanation
480
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
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 explanation
357
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
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 explanation
572
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
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 explanation
437
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
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 explanation
672
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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 explanation
none
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
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 explanation
1
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
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 explanation
2
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
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 explanation
none
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
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 explanation
1
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
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 explanation
2
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
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 explanation
none
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
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 explanation
1
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
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 explanation
2
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
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 explanation
none
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
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 explanation
1
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
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 explanation
2
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
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 explanation
none
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
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 explanation
1
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
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 explanation
2
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
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 explanation
none
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
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 explanation
1
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
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 explanation
2
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
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 explanation
none
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
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 explanation
1
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
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 explanation
2
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
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 explanation
none
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
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 explanation
1
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
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 explanation
2
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
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 explanation
none
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
Real eigenvalues and nonzero eigenvectors of 2×2 matrices · qf-density-eigen-first-mismatch · candidate
Canonical question JSONL · English JSONL · Validation report