Question

A and B are square matrices of the same order 3, such that AB = 2I and |A| = 2, write the value of |B|.

Answer 1

Given:
A and B are square matrices of order 3
|A| = 2
AB = 2I


$$\begin{gathered} \mathrm{Now}, \\ AB=2I \\ \Rightarrow \left|AB\right|=\left|2I\right| \\ \Rightarrow \left|A\right|\left|B\right|=\left|2I\right|\left(\because \left|AB\right|=\left|A\right|\left|B\right|,\mathrm{if}\mathrm{they}\mathrm{are}\mathrm{square}\mathrm{matrices}\mathrm{of}\mathrm{same}\mathrm{order}\right) \\ \Rightarrow 2\left|B\right|=\left|2I\right|\left(\because \left|A\right|=2\right) \\ \Rightarrow 2\left|B\right|=2^3\left|I\right|\left(\because \mathrm{Order}\mathrm{of}I\mathrm{is}3\times 3\right) \\ \Rightarrow 2\left|B\right|=8\times 1\left(\because \left|I\right|=1\right) \\ \Rightarrow \left|B\right|=4 \end{gathered}$$

Hence, the value of |B| is 4.