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 :
(4) 16
R2-2R1: y+(a-4)z=a-11
R3+R1: -2y+(b+2)z=2b+6
Eliminate y: (2a+b-6) z=2a+2b-16
Infinitely many solutions both sides =0
2a+b=6, 2a+2b=16
b=10, a=-2
7a+3b=-14+30=16
Answer: (4)
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 :
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 :
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 :
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 :
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
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 :
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 :