Matrices · Question 10

For some , R, let A = bmatrix & 2 \ 1 & 2 bmatrix and B = bmatrix 1 & 1 \ 1 & bmatrix be such that A2 - 4A + 2I = B2 - 3B + I = O. Then ( ( adj(A3 - B3)))2 is equal to .

For the matrices A = bmatrix 3 & -4 \ 1 & -1 bmatrix and B = bmatrix -29 & 49 \ -13 & 18 bmatrix, if (A15 + B) bmatrix x \ y bmatrix = bmatrix 0 \ 0 bmatrix, then among the following which one is true?

1. x = 16, y = 3
2. x = 5, y = 7
3. x = 11, y = 2
4. x = 18, y = 11

Let A = bmatrix & 0 & - \ 0 & 1 & 0 \ & 0 & bmatrix. If for some (0, ), A2 = AT, then the sum of the diagonal elements of the matrix (A + I)3 + (A - I)3 - 6A is equal to .

Let the matrix A = bmatrix 1 & 0 & 0 \ 1 & 0 & 1 \ 0 & 1 & 0 bmatrix satisfy An = An-2 + A2 - I for n 3. Then the sum of all the elements of A50 is :

1. 44
2. 39
3. 52
4. 53

Let A = bmatrix & -1 \ 6 & bmatrix, > 0, such that (A) = 0 and + = 1. If I denotes 2 2 identity matrix, then the matrix (I + A)8 is :

1. bmatrix 257 & -64 \ 514 & -127 bmatrix
2. bmatrix 766 & -255 \ 1530 & -509 bmatrix
3. bmatrix 1025 & -511 \ 2024 & -1024 bmatrix
4. bmatrix 4 & -1 \ 6 & -1 bmatrix

Let A be a 3 3 real matrix such that A2(A - 2I) - 4(A - I) = O, where I and O are the identity and null matrices, respectively. If A5 = A2 + A + I, where , , and are real constants, then + + is equal to :

1. 76
2. 12
3. 4
4. 20

Let S = \m Z : Am2 + Am = 3I - A-6 \, where A = bmatrix 2 & -1 \ 1 & 0 bmatrix. Then n(S) is equal to .

Let M denote the set of all real matrices of order 3 3 and let S = \-3, -2, -1, 1, 2\. Let S1 = \A = [aij] M : A = AT and aij S, i, j\, S2 = \A = [aij] M : A = -AT and aij S, i, j\, S3 = \A = [aij] M : a11 + a22 + a33 = 0 and aij S, i, j\. If n(S1 S2 S3) = 125 , then equals .

Let A = bmatrix 1 2 & -2 \ 0 & 1 bmatrix and P = bmatrix & - \ & bmatrix, > 0. If B = PAP , C = P B10P and the sum of the diagonal elements of C is mn, where (m, n) = 1, then m + n is :

1. 127
2. 2049
3. 258
4. 65