Question

If k is a scalar and I is a unit matrix of order 3, then adj (kI) = ________________.

Answer 1

Given:
I is a unit matrix of order 3


As we know,
$$\begin{gathered} A\left(\mathrm{adj}A\right)=\left|A\right|I \\ \Rightarrow A^{-1}A\left(\mathrm{adj}A\right)=A^{-1}\left|A\right|I \\ \Rightarrow \left(A^{-1}A\right)\left(\mathrm{adj}A\right)=A^{-1}\left|A\right|I \\ \Rightarrow I\left(\mathrm{adj}A\right)=\left|A\right|\left(A^{-1}I\right) \\ \Rightarrow \mathrm{adj}A=\left|A\right|A^{-1} \\ \Rightarrow \mathrm{adj}\left(kI\right)=\left|kI\right|{\left(kI\right)}^{-1} \\ \Rightarrow \mathrm{adj}\left(kI\right)=k^3\left|I\right|{\left(kI\right)}^{-1}\left(\because \mathrm{order}\mathrm{of}I\mathrm{is}3\right) \\ \Rightarrow \mathrm{adj}\left(kI\right)=k^3\times 1\times {\left(kI\right)}^{-1} \\ \Rightarrow \mathrm{adj}\left(kI\right)=k^3{k}^{-1}{\left(I\right)}^{-1} \\ \Rightarrow \mathrm{adj}\left(kI\right)=k^2{\left(I\right)}^{-1} \\ \Rightarrow \mathrm{adj}\left(kI\right)=k^{2}I\left(\because I^{-1}=I\right) \\ \Rightarrow \mathrm{adj}\left(kI\right)=k^{2}I \end{gathered}$$


Hence, adj (kI) = k²I.