Matrices

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 A = [aij] be a matrix of order 3 3, with aij = ( 2)i+j. If the sum of all the elements in the third row of A2 is + 2, , Z, then + is equal to :

1. 210
2. 280
3. 224
4. 168

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

Let B = bmatrix 1 & 3 \ 1 & 5 bmatrix and A be a 2 2 matrix such that AB-1 = A-1. If BCB-1 = A and C4 + C2 + I = O, then 2 - is equal to

1. 16
2. 10
3. 8
4. 2

Let A = bmatrix 2 & a & 0 \ 1 & 3 & 1 \ 0 & 5 & b bmatrix. If A3 = 4A2 - A - 21I, where I is the identity matrix of order 3 3, then 2a + 3b is equal to

1. -10
2. -12
3. -13
4. -9