Question

If $A=\left[\begin{array}{cc}2& 3\\ 5& -2\end{array}\right]$ be such that $A^{-1}=kA,$ then find the value of k.

Answer 1

$$\begin{gathered} A=\left[23 \\ 5-2\right] \\ \therefore \left|A\right|=\left|23 \\ 5-2\right|=-14-15=-19 \\ \mathrm{The}\mathrm{value}\mathrm{is}\mathrm{non}-\mathrm{zero},\mathrm{so}{A}^{-1}\mathrm{exists}. \\ \mathrm{By}\mathrm{definition},\mathrm{we}\mathrm{have} \\ {A}^{-1}A=I[I\mathrm{is}\mathrm{the}\mathrm{identity}\mathrm{matrix}] \\ \Rightarrow kA.A=I[\mathrm{Substituting}{A}^{-1}=kA] \\ \Rightarrow k\left[23 \\ 5-2\right]\left[23 \\ 5-2\right]=\left[10 \\ 01\right] \\ \Rightarrow k\left[4+156-6 \\ 10-1015+4\right]=\left[10 \\ 01\right] \\ \Rightarrow k\left[190 \\ 019\right]=\left[10 \\ 01\right] \\ \Rightarrow k=\frac{1}{19} \end{gathered}$$