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Identity Matrix

If A and B ar...

Question

If A and B are two non-singular matrices then $| A d j (A B) | =$

A

| A d j (B) | | A d j A |

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B

| A d j A | | A d j B |

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C

Both

(1)

and

(2)

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D

| A | | B |

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Solution

The correct option is D Both $(1)$ and $(2)$
$| A B | = | A | | B |$
$(A B) (a d j . A B) = | A B | I_{n}$
$(A B) (a d j . B . a d j . A) = | A B / I_{n} |$
$= A (B . a d j . B) a d j . A$
$= (A | B / I_{n} | a d j . A)$
$= | B | (A . a d j . A)$
$= | B | | A |$
$(A . a d j . A) = | A |$ & $(B . a d j . B) = | B |$
$∴ (A B) (a d j (A B)) = (B . a d j . B) (A . a d j . A)$
$∴ (A B) (a d j . A B) = (A B) (a d j . B . a d j . A)$
$∴ a d j . A B = a d j . B . a d j . A$ .

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Similar questions

Q. Let A, B are two non-singular matrices of order 3 with real entries such that

adj (A) = 3 B

adj (B) = 2 A

A = ⎡ ⎢ ⎣ \begin{matrix} 1 & - 1 & 1 0 & 2 & - 3 2 & 1 & 0 \end{matrix} ⎤ ⎥ ⎦

B = (a d j A)

C = 5 A

\frac{| a d j B |}{| C |} =

A = ∣ ∣ ∣ ∣ \begin{matrix} 1 & - 1 & 1 0 & 2 & - 3 2 & 1 & 0 \end{matrix} ∣ ∣ ∣ ∣

B = (a d j A)

C = 5 A

\frac{| a d j B |}{| C |}

A = ⎡ ⎢ ⎣ \begin{matrix} 1 & - 1 & 1 0 & 2 & - 3 2 & 1 & 0 \end{matrix} ⎤ ⎥ ⎦, B = (a d j A)

C = 5 A

\frac{| a d j B |}{| C |}

A = ⎡ ⎢ ⎣ \begin{matrix} 1 & - 1 & 1 0 & 2 & - 3 2 & 1 & 0 \end{matrix} ⎤ ⎥ ⎦, B = (a d j A)

C = 5 A

\frac{| C |}{| a d j B |}

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