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Inverse of a Matrix

If A and ...

Question

If $A$ and $B$ are invertible matrices of order $3, | A | = 2$ and $∣ ∣ {(A B)}^{- 1} ∣ ∣ = - \frac{1}{6}$ Find $| B |$

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Solution

A & B are invertible matrices
$∴ | A |$ or $| B | \neq 0$
and $| A | = 2$
$∣ ∣ (A B)^{- 1} ∣ ∣ = \frac{- 1}{6}; | B | = ?$
since $∣ ∣ (A B)^{- 1} ∣ ∣ = \frac{- 1}{6}$
$∴ ∣ ∣ B^{- 1} A^{- 1} ∣ ∣ = \frac{- 1}{6}$
$∣ ∣ B^{- 1} ∣ ∣ ∣ ∣ A^{- 1} ∣ ∣ = \frac{- 1}{6}$
$[(A B)^{- 1} = B^{- 1} A^{- 1} (∣ ∣ A^{- 1} ∣ ∣ = \frac{1}{A})$
$(∣ ∣ B^{- 1} A^{- 1} ∣ ∣ = ∣ ∣ B^{- 1} ∣ ∣ ∣ ∣ A^{- 1} ∣ ∣)]$
$∣ ∣ B^{- 1} ∣ ∣ \frac{1}{2} = \frac{- 1}{6} ∴ ∣ ∣ B^{- 1} ∣ ∣ = \frac{- 1}{3}$
$∴ | B | = \frac{1}{| B^{- 1} |} = - 3$
$∴ | B | = - 3$

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Similar questions

Q. If A and B are invertible matrices of order

3

| A | = 2

∣ ∣ {(A B)}^{- 1} ∣ ∣ = - \frac{1}{6}

| B |

Q. If A and B are invertible matrices of order 2, |A| =2 and

| (A B)^{- 1} | = - \frac{1}{6}

Q. If A and B are invertible matrices of order 3,

| A | = 2

{| (A B) |}^{- 1} | = - \frac{1}{6}

A

B

are invertible matrices of the same order, then prove that

(A B)^{- 1} = B^{- 1} A^{- 1}

Q. If A is an invertible matrix of order 3, then which of the following is not true
(a)

|adj A| = {|A|}^{2}

{(A^{- 1})}^{- 1} = A

B A = C A, than B \neq C

, where B and C are square matrices of order 3
(d)

{(A B)}^{- 1} = B^{- 1} A^{- 1}, where B \neq {[b_{i j}]}_{3 \times 3} and |B| \neq 0

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