Question

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

Answer 1

As we know, $|A{A}^{-1}|=1$

${|A}^{-1}|=\frac{1}{|A|}=\frac{1}{2}$ (Since $|A|=2$given)

${\left(AB\right)}^{-1}=\frac{-1}{6}$

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

From the value of $A^{-1}$ above we get,

$B^{-1}\frac{1}{2}=\frac{-1}{6}$

$B^{-1}=\frac{-2}{6}=\frac{-1}{3}$

$B^{-1}=\frac{-1}{3}$

$|B{B}^{-1}|=1$

$|B|=\frac{1}{|B^{-1}|}=-3$

$|B|=-3$