Matrices

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

Let A be a 3 3 matrix such that XTAX=O for all nonzero 3 1 matrices X= bmatrixx\ \ bmatrix. If A bmatrix1\\1\\1 bmatrix= bmatrix1\\4\\-5 bmatrix, A bmatrix1\\2\\1 bmatrix= bmatrix0\\4\\-8 bmatrix, and (adj(2(A+I)))=2 3 5 , , , N, then 2+ 2+ 2 is

Let A=[aij] be a square matrix of order 2 with entries either 0 or 1. Let E be the event that A is an invertible matrix. Then the probability P(E) is :

1. 38
2. 18
3. 316
4. 58

Let A=[aij] be a 3 3 matrix such that A bmatrix0\\1\\0 bmatrix= bmatrix0\\0\\1 bmatrix, A bmatrix4\\1\\3 bmatrix= bmatrix0\\1\\0 bmatrix and A bmatrix2\\1\\2 bmatrix= bmatrix1\\0\\0 bmatrix, then a23 equals :

1. 2
2. -1
3. 1
4. 0