12 There exists a real number k such that the matrix 1 k 4 k 2 fails to be

12 there exists a real number k such that the matrix

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12. There exists a real number k such that the matrix 1 [ k -=-4 k -=-2 ] fails to be invertible. 3 13. There exists a real number k such that the matrix 2 [ k -=-3 k 2] fails to be invertible. I 1/2]. I . . . 14. A = 0 1/2 IS a regu ar transItion matnx. [ 15. The fonnula det(2A) = 2 det( A) holds for all 2 x 2 matrices A. 16. There exists a matrix A such that 17. Matrix [3 1 26] is invertible. 18. Matrix is invertible. 19. There exists an upper triangular 2 x 2 matrix A such 2 that A = ] . 20. The function T [:] = = is a linear transformation. 21. There exists an invertible n x n matrix with two identi- cal rows. 22. If A 2 = In, then matrix A must be invertible. 23. There exists a matrix A such that A 1 11] __ [11 2] 2 . [ 24. There exists a matrix A such that A = ] . 25. The matrix [I 1] represents a reflection about a I -I line. 26. For every regular transition matrix A there exists a tran- sition matrix B such that AB = B. h · [a b] [d -b]. I 27. T e matnx product c d -c a IS a ways a scalar multiple of h. 28. There exists a nonzero upper triangular 2 x 2 matrix A suchthatA 2 = 29. There exists a positive integer n such that [ 0 -I] n 1 ° = h 30. There exists an invertible 2 x 2 matrix A such that A-I = 31. There exists a regular transition matrix A of size 3 x 3 such that A 2 = A. 32. If A is any transition matrix and B is any positive tran- sition matrix, then AB must be a positive transition ma- trix. 33. If matrix ag hbe Jet"] IS invertible, then matrix [d a be] must be invertible as well. [d 34. If A 2 is invertible, then matrix A itself must be invert- ible. 35. If A 17 = h, then matrix A must be h. 36. If A 2 = h, then matrix A must be either h or -h. 37. If matrix A is invertible, then matrix SA must be invert- ible as well. 38. If A and B are two 4 x 3 matrices such that Av = Bv for all vectors v in ]R3, then matrices A and B must be equal. 39. If matrices A and B commute, then the formula A 2 B = B A 2 must hold. 40. If A 2 = A for an invertible n x n matrix A, then A must be In. 41. If A is any transition matrix such that A 100 is positive, then A 101 must be positive as well. 42. If a transition matrix A is invertible, then A -I must be a transition matrix as well. 43. If matrices A and B are both invertible, then matrix A + B must be invertible as well. 44. The equation A 2 = A holds for all 2 x 2 matrices A representing a projection. 45. The equation A -I = A holds for all 2 x 2 matrices A representing a reflection. 46. The formula (Av)· (Aw) = w holds for all invertible 2 x 2 matrices A and for all vectors v and w in ]R2. 47. There exist a 2 x 3 matrix A and a 3 x 2 matrix B such that AB = h 48. There exist a 3 x 2 matrix A and a 2 x 3 matrix B such that AB = h 49. If A 2 + 3A +413 = 0 for a 3 x 3 matrix A, then A must be invertible. 50. If A is an n x n matrix such that A 2 = 0, then matrix In + A must be invertible.
51. If matrix A commutes with B, and B commutes with C, then matrix A must commute with C.

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