Matrices

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

Consider the matrix f(x)= bmatrix x & - x & 0\ x & x & 0\ 0 & 0 & 1 bmatrix. Given below are two statements : In the light of the above statements, choose the correct answer from the options given below :

1. Statement I is false but Statement II is true
2. Both Statement I and Statement II are false
3. Both Statement I and Statement II are true
4. Statement I is true but Statement II is false

Let A be a 2 2 real matrix and I be the identity matrix of order 2. If the roots of the equation |A-xI|=0 be -1 and 3, then the sum of the diagonal elements of the matrix A2 is .