Matrices

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

Let A = bmatrix 2 & -1 \ 1 & 1 bmatrix. If the sum of the diagonal elements of A13 is 3n, then n is equal to .

Let A be a square matrix of order 2 such that |A|=2 and the sum of its diagonal elements is -3. If the points (x, y) satisfying A2+xA+yI=O lie on a hyperbola, whose transverse axis is parallel to the x-axis, eccentricity is e and the length of the latus rectum is l, then e4+l4 is equal to .

Let A = bmatrix1 & 2\\0 & 1 bmatrix and B = I + adj(A) + (adj A)2 + + (adj A)10. Then, the sum of all the elements of the matrix B is:

1. -110
2. 22
3. -124
4. -88

Let A be a 2 2 symmetric matrix such that A bmatrix1\\1 bmatrix = bmatrix3\\7 bmatrix and the determinant of A be 1. If A-1 = A + I, where I is an identity matrix of order 2 2, then + equals .

Let A=I2-2MMT, where M is a real matrix of order 2 1 such that the relation MTM=I1 holds. If is a real number such that the relation AX= X holds for some non-zero real matrix X of order 2 1, then the sum of squares of all possible values of is equal to .

Let A be a square matrix such that AAT=I. Then 12A [ (A+AT )2+ (A-AT )2 ] is equal to

1. A2+AT
2. A3+I
3. A3+AT
4. A2+I