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Orthogonal Matrix

If A and B ar...

Question

If A and B are two invertible matrices of the same order, then adj(AB) is equal to

A

adj(B) adj(A)

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B

|B| |A| B

^{- 1}

A

^{- 1}

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C

|B| |A| A

^{- 1}

B

^{- 1}

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D

|A| |B| (AB)

^{- 1}

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Solution

The correct options are
A adj(B) adj(A)
B |B| |A| B $^{- 1}$ A $^{- 1}$
D |A| |B| (AB) $^{- 1}$
We have $a d j A = | A | A^{- 1}$
$∴ a d j (A B) = | A B | (A B)^{- 1} = | A | | B | (B^{- 1} A^{- 1})$
$\Rightarrow (| B | B^{- 1}) (| A | A^{- 1}) = (a d j B) (a d j A)$

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