Determinants

For , R and a natural number n, let Ar = vmatrix r & 1 & n22 + \ 2r & 2 & n2 - \ 3r - 2 & 3 & n(3n - 1)2 vmatrix. Then 2A10 - A8 is

1. 4 + 2
2. 0
3. 2n
4. 2 + 4

Let = 45; , , R. If x( , 1, 2) + y(1, , 2) + z(2, 3, ) = (0, 0, 0) for some x, y, z R, xyz 0, then 6 + 4 + is equal to .

If A is a square matrix of order 3 such that (A) = 3 and (adj(-4 adj(-3 adj(3 adj((2A)-1))))) = 2m3n, then m + 2n is equal to :

1. 2
2. 4
3. 3
4. 6

If the system of equations 2x + 7y + z = 3 3x + 2y + 5z = 4 x + y + 32z = -1 has infinitely many solutions, then ( - ) is equal to :

Let A and B be two square matrices of order 3 such that |A| = 3 and |B| = 2. Then |ATA(adj(2 A))-1(adj(4 B))(adj(AB))-1AAT| is equal to :

1. 32
2. 81
3. 64
4. 108

If the system of equations 11x + y + z = -5 2x + 3y + 5z = 3 8x - 19y - 39z = has infinitely many solutions, then 4 - is equal to :

1. 51
2. 45
3. 47
4. 49

The values of m, n, for which the system of equations x + y + z = 4, 2x + 5y + 5z = 17, x + 2y + mz = n has infinitely many solutions, satisfy the equation :

1. m2 + n2 - m - n = 46
2. m2 + n2 + mn = 68
3. m2 + n2 - mn = 39
4. m2 + n2 + m + n = 64

Let 0 and A = bmatrix & & 3 \ & & \ - & & 2 bmatrix. If B = bmatrix 3 & -9 & 3 \ - & 7 & -2 \ -2 & 5 & -2 bmatrix is the matrix of cofactors of the elements of A, then (AB) is equal to :

1. 64
2. 343
3. 125
4. 216

Let (0, ) and A= bmatrix1 & 2 & \ 1 & 0 & 1 \ 0 & 1 & 2 bmatrix. If det ( adj (2A-AT ) adj (A-2AT ) )=28, then ( det(A))2 is equal to:

1. 16
2. 36
3. 49
4. 1

If the system of equations x+( 2 )y+( 2 )z=0 x+( )y+( )z=0 x+( )y-( )z=0 has a non-trivial solution, then (0, 2 ) is equal to :

1. 5 24
2. 11 24
3. 7 24
4. 3 4