Match the following columns: Column I (A,B,C are matrices) Column-II

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Solution

Q.
Match the following columns:

Column I (A,B,C are matrices) Column-II

a. If $| A_{3 \times 3} |$ $= 2$ , then $| 2 A^{- 1} |$ $=$ p. 1

b. If $| A_{3 \times 3} |$ $= \frac{1}{8}$ , then $| a d j (a d j (2 A)) |$ $=$ q. 4

c. If ${(A + B)}^{2} = A^{2} + B^{2}$ , and $| A_{2 \times 2} |$ $= 2$ , then $| B_{2 \times 2} |$ $=$ r. 8

d. $| A_{2 \times 2} |$ $= 2,$ $| B_{3 \times 3} |$ $= 3$ and $| C_{4 \times 4} |$ $= 4$ , then $| A B C |$ is equal to s. 0

t. Does not exist

1 a-q; b-p; c-s; d-t

2 a-p; b-r; c-q; d-s

3 a-p; b-q; c-r; d-s

4 a-s; b-q; c-r; d-p

Ans.
(1)

a. $| A |$ $= 2 \Rightarrow$ $| 2 A^{- 1} |$ $= 2^{3} /$ $| A |$ $= 4$

b. $| a d j (a d j (2 A)) |$ $=$ $| 2 A |^{4}$ $= 2^{12}$ $| A |^{4}$ $= 2^{12} / 2^{12} = 1$

c. ${(A + B)}^{2} = A^{2} + B^{2} \Rightarrow A B + B A = O$

$\Rightarrow$ $| A B |$ $=$ $| - B A |$ $= -$ $| B A |$ $= -$ $| A B |$ $\Rightarrow$ $| A B |$ $= 0 \Rightarrow$ $| B |$ $= 0$

d. Product ABC is not defined.

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