If A and B are invertible matrices of order 3, | A | = 2 and | ( A B ) - 1 | = - 1 6 , then the value of | B | is:

Question

If $A$ and $B$ are invertible matrices of order 3, $| A |$ $= 2$ and $| {(A B)}^{- 1} |$ $= - \frac{1}{6},$ then the value of $| B |$ is:

Smart Achievers · Accepted Answer

(2)

Ans. $- 3$

Explanation:

Given, $| A |$ $= 2,$

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

We know that

$| B B^{- 1} |$ $=$ $I$ and $| A A^{- 1} |$ $= I$

$| B^{- 1} |$ $= \frac{1}{B}$ and $| A^{- 1} | = \frac{1}{| A |} = \frac{1}{2}$

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

$\Rightarrow$ $| B^{- 1} |$ $| A^{- 1} |$ $= \frac{- 1}{6}$

$\Rightarrow$ $\frac{1}{B} \times \frac{1}{2} = \frac{- 1}{6}$

$\Rightarrow$ $B = - 3$

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