Matrices

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) A^{n - 1}       3) -Aⁿ     4) -A^{n - 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 + ....... + A^{n - 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. M_r = 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 = -A^T ; B = -B^T
(AB)^T = B^TA^T
           = (-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 = 3^{n - 1}
 35 = 3^{n - 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 = A^m . Bⁿ, then m × n =
1) 13020        2) 12300       3) 10320          4) 10230
Ans: (4)