Determinants

The system of equations x+y+z=6, x+2y+5z=9, x+5y+ z= , has no solution if

1. =17, =18
2. =17, 18
3. =15, 17
4. 17, 18

Let A be a square matrix of order 3 such that (A)=-2 and (3 adj (-6 adj(3A) ) )=2m+n 3mn, m>n. Then 4m+2n is equal to .

If the system of linear equations : x+y+2z=6 2x+3y+az=a+1 -x-3y+bz=2b where a,b , has infinitely many solutions, then 7a+3b is equal to :

1. 12
2. 9
3. 22
4. 16

For a 3 3 matrix M, let trace (M) denote the sum of all the diagonal elements of M. Let A be a 3 3 matrix such that |A|= 12 and trace (A)=3. If B=adj(adj(2A)), then the value of |B|+ trace (B) equals :

1. 56
2. 132
3. 174
4. 280

Let , R. If the system of equations 3x + 5y + z = 3 7x + 11y - 9z = 2 97x + 155y - 189z = has infinitely many solutions, then + 2 is equal to :

1. 24
2. 25
3. 27
4. 22

Let A be a non-singular matrix of order 3. If (3 adj(2 adj(( A)A))) = 3-13 2-10 and (3 adj(2A)) = 2m 3n, then |3m + 2n| is equal to .

Consider the matrices : A = bmatrix 2 & -5 \ 3 & m bmatrix, B = bmatrix 20 \ m bmatrix and X = bmatrix x \ y bmatrix. Let the set of all m, for which the system of equations AX = B has a negative solution (i.e., x < 0 and y < 0), be the interval (a, b). Then 8 ab |A| dm is equal to .

If a, b, c and vmatrix & b & c \ a & & c \ a & b & vmatrix = 0, then a - a + b - b + - c is equal to :

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

If the system of equations x + 4y - z = , 7x + 9y + z = -3, 5x + y + 2z = -1 has infinitely many solutions, then (2 + 3 ) is equal to :

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

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