Matrices

Consider the matrices A= bmatrix2 & -2\ 4 & -2 bmatrix and B= bmatrix3 & 9\ 1 & 3 bmatrix. If matrices P and Q are such that PA=B and AQ=B, then the absolute value of the sum of the diagonal elements of 2(P+Q) is .

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

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 .

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 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 .