Matrices and Determinants Questions and Answers Part-1

1. Let $A=\left(\begin{array}{c}a & b\\ c & d\end{array}\right)$    be such that $A^{3}=O$  , but $A\neq O$  , then
a) $A^{2}=O$
b) $A^{2}=A$
c) $A^{2}=I-A$
d) none of these

Answer: a
Explanation: As A3 = O, we get |A3| = 0

2. Let S be the set of all real matrices $A =\begin{bmatrix}a & b \\c & d \end{bmatrix}$    such that a + d = 3 and $A'=A^{2}-3A$   . Then S:
a) is an empty set
b) has exactly one element
c) has exactly two elements
d) has exactly four elements

Answer: a
Explanation:

3. Let $A =\begin{bmatrix}a & b \\c & d \end{bmatrix}$   , be a 2 × 2 matrix where a, b, c, $d \epsilon$ {0, 1}. The number of such matrices which have inverse is
a) 5
b) 6
c) 7
d) 8

Answer: b
Explanation: det(A) = ad – bc

4. Let $A =\begin{bmatrix}a & b \\c & d \end{bmatrix}$   , where a, b, c, d $\epsilon R$ . If $A -\alpha I$   is invertible for all a $\alpha \epsilon$ R, then
a) bc > 0
b) bc=0
c) $bc> min\left(0,\frac{1}{2}ad\right)$
d) a=0

Answer: c
Explanation: As A – $\alpha$ I is invertible for all a $\epsilon$ R

5. If A + B is a non-singular matrix, then $A – B – A \left(A + B\right)^{-1}A + B(A + B)^{-1} B$
equals
a) O
b) I
c) A
d) B

Answer: a
Explanation:

6. If P is a 3 * 3 matrix such that P' = 2P + I, then there exists a column matrix $X=\left(\begin{array}{c}x\\ y \\ z \end{array}\right)\neq O$
such that
a) PX = O
b) PX = X
c) PX = 2X
d) PX = -X

Answer: d
Explanation:

7. Let P and Q be $3\times 3$  matrices with $P\neq Q$   If $P^{3}=Q^{3}$  and $P^{2}Q=Q^{2}P$   , then determinant of $\left(P^{2}+Q^{2}\right)$    is equal to
a) 1
b) 0
c) -1
d) -2

Answer: b
Explanation:

8. Let $P=\begin{bmatrix}1 & 0 & 0 \\9 & 1 & 0 \\27 & 9 & 1 \end{bmatrix}$     and $Q=\left[q_{ij}\right]_{3\times 3}$    be such that $P^{5}-Q=I$  , then $\frac{q_{21}+q_{31}}{q_{32}}$   is equal to
a) 22
b) 33
c) 44
d) 55

Answer: a
Explanation: Write P = I + R where

9. Let $a_{ij}=\left(2+\sqrt{3}\right)^{i+j}+\left(2-\sqrt{3}\right)^{i+j},1 \leq i, j \leq3$         and let $A=\left(a_{ij}\right)_{3\times 3}$     , then det (A) is equal to
a) 1
b) $\left(2+\sqrt{3}\right)^{9}$
c) $\left(2-\sqrt{3}\right)^{9}$
d) 0

Answer: d
Explanation: For n $\epsilon$ N

10. If $A_{\alpha}=\begin{bmatrix}\cos \alpha & \sin\alpha \\-\sin\alpha & \cos\alpha \end{bmatrix}$     , then $A_{\alpha}A_{\beta}$  is equal to
a) $A_{\alpha+\beta}$
b) $A_{\alpha\beta}$
c) $A_{\alpha-\beta}$
d) none of these

Answer: a
Explanation: