Matrices

Let A = bmatrix 3 & -4 \ 1 & -1 bmatrix and B be two matrices such that A100 = 100B + I. Then the sum of all the elements of B100 is

The number of 3 2 matrices A, which can be formed using the elements of the set \-2, -1, 0, 1, 2\ such that the sum of all the diagonal elements of ATA is 5, is

Let A = bmatrix 0 & 2 & -3 \ -2 & 0 & 1 \ 3 & -1 & 0 bmatrix and B be a matrix such that B(I - A) = I + A. Then the sum of the diagonal elements of BTB is equal to

If A = bmatrix 2 & 3 \ 3 & 5 bmatrix, then the determinant of the matrix (A2025 - 3 A2024 + A2023 ) is

1. 12
2. 24
3. 28
4. 16

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