1. Let A be a 3 × 3 matrix and B, its adjoint matrix. If det. B = 64, then det. A =
1) ± 4 2) ± 8 3) ± 12 4) ± 24
Ans: (2)
Explanation: | Adj. A| = |A|n - 1 = det. B
64 = |A|3 - 1 = |A|2
b |A| = ± 8.
2. A is a square matrix satisfying the equation A2 - 4A - 5I = 0. Then A-1 =
3. A non - singular matrix A is such that I + A + A2 + ..... + An = 0, where O is a null matrix of the same order as A and I. Then A-1 =
1) An 2) An - 1 3) -An 4) -An - 1
Ans: (1)
Explanation: A is non - singular.
... A-1 exists.
I + A + A2 + ....... + An = 0
A-1(I + A + A2 + ....... + An) = A-1 O
A-1 + I + A + A2 + ....... + An - 1 = 0
A-1 + O - An = 0 (By data)
A-1 = An
4. If the matrix Mr is given by
1) 2012 2) 2013 3) (2012)2 4) (2013)2
Ans: (4)
Explanation: det. Mr = r2 - (r - 1)2 = 2r - 1
= 1 + 3 + 5 + 7 + ...... + 4025
= (2013)2 [ 1 + 3 + 5 + ...... n terms = n2]
7. A and B are each a skew - symmetric matrix of order 3 × 3. If AB = BA, then AB is a
1) scalar matrix 2) symmetric matrix
3) unit matrix 4) skew - symmetric matrix
Ans: (2)
Explanation: A = -AT ; B = -BT
(AB)T = BTAT
= (-B)(-A)
= BA
= AB
... AB is a symmetric matrix.
8. A square matrix and its adjoint have their determinant values as 3 and 243 respectively. The order of the matrix is
1) 4 × 4 2) 5 × 5 3) 6 × 6 4) 3 × 3
Ans: (3)
Explanation: det. A = 3; det. (adj. A) = 343
det. (adj. A) = (det. A)n - 1 where n represents the order of the matrix.
343 = 3n - 1
35 = 3n - 1 ... n = 6
... Order of the matrix is 6 × 6.
11. If A and B are square matrices of the same order and B = -A-1BA, then
1) AB = -BA 2) AB = BA
3) A2 = B2 4) A2 = (A + B)2 - B2
Ans: (1)
Explanation: B = -(A-1 BA)
AB = -(AA-1 BA)
= - (IBA)
= - BA
13. A is a matrix of order x(x + 5) whereas B is another matrix of order is y(11 - y). If both are conformable for multiplication such that AB = BA, then their orders respectively are
1) 2 × 5, 5 × 2 2) 2 × 6, 6 × 2 3) 3 × 7, 7 × 3 4) 3 × 8, 8 × 3
Ans: (4)
Explanation: AB x + 5 = y
BA 11 − y = x
Solving, x = 3, y = 8
... Orders of A, B are 3 × 8 and 8 × 3 respectively.
15. Two square matrices A and B are such that A2B = BA. If (AB)10 = Am . Bn, then m × n =
1) 13020 2) 12300 3) 10320 4) 10230
Ans: (4)