Let $A$ and $B$ be non singular square matrices of order $3$ satisfying $A + a d j A = B$ and $| A | = 3,$ then $| (a d j A) B - 3 I |$ is equal to:

9

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27

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81

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243

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Solution

The correct option is C $81$
$A + a d j A = B$
Multiply both sides by $(a d j A)$
$(a d j A) A + (a d j A)^{2} = (a d j A) B | A | I + (a d j A)^{2} = (a d j A) B (∵ a d j A = | A | A^{- 1})$
$| (a d j A) B - 3 I | = | (a d j A) |^{2} = | A |^{4} = 81$

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Similar questions

Let A be the non-singular square matrix of order $3 \times 3$ then |adj A| is equal to

a) $| A |$
b) $| A |^{2}$
c) $| A |^{3}$
d) $3 | A |$

A

B

A B - B A = A

adj (A) = 3 B

adj (B) = 2 A

| A d j (A B) | =

A^{3} = B^{3}

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