Question

Let A be a square matrix of order 3 and B = $\left|A\right|A^{-1}.\mathrm{If}\left|\mathrm{A}\right|=-5,\mathrm{then}\left|\mathrm{B}\right|$ = _________________.

Answer 1

Given:
A is a square matrix of order 3
B = |A|A⁻¹
|A| = −5


Now,
$$\begin{gathered} B=\left|A\right|A^{-1} \\ \Rightarrow \left|B\right|=\left|\left|A\right|A^{-1}\right| \\ \Rightarrow \left|B\right|={\left|A\right|}^3\left|A^{-1}\right|\left(\because \mathrm{order}\mathrm{of}A\mathrm{is}3\right) \\ \Rightarrow \left|B\right|={\left|A\right|}^3\frac{1}{\left|A\right|}\left(\because \left|A^{-1}\right|=\frac{1}{\left|A\right|}\right) \\ \Rightarrow \left|B\right|={\left|A\right|}^2 \\ \Rightarrow \left|B\right|={\left(-5\right)}^2\left(\because \left|A\right|=-5\right) \\ \Rightarrow \left|B\right|=25 \end{gathered}$$


Hence, $\left|B\right|=\underline{25}$.