Inverse of a diagonal non-singular matrix is

Let diagonal matrix \(A=\left[\begin{array}{lll}a & 0 & 0 \\ 0 & a & 0 \\ 0 & 0 & a\end{array}\right]\)
Now, \(\quad|\mathrm{A}|=\mathrm{a}\left(\mathrm{a}^{2}\right)=\mathrm{a}^{3}\)
\(\therefore\)
\(\operatorname{adj} A=\left[\begin{array}{ccc}a^{2} & 0 & 0 \\ 0 & a^{2} & 0 \\ 0 & 0 & a^{2}\end{array}\right]\)
\(\begin{aligned} \therefore \mathrm{A}^{-1}=\frac{\operatorname{adj}(\mathrm{A})}{|\mathrm{A}|} &=\frac{1}{\mathrm{a}^{3}}\left[\begin{array}{ccc}\mathrm{a}^{2} & 0 & 0 \\ 0 & \mathrm{a}^{2} & 0 \\ 0 & 0 & \mathrm{a}^{2}\end{array}\right] \\ &=\left[\begin{array}{ccc}1 / \mathrm{a} & 0 & 0 \\ 0 & 1 / \mathrm{a} & 0 \\ 0 & 0 & 1 / \mathrm{a}\end{array}\right] \end{aligned}\)
which is an diagonal matrix.