Question

If $A=\left[\begin{array}{cc}1& -3\\ 2& 0\end{array}\right]$, write adj A.

Answer 1

$$\begin{gathered} \left|A\right|=\left|\begin{array}{cc}1& -3\\ 2& 0\end{array}\right|=6\ne 0 \\ A\mathrm{is}\mathrm{a}\mathrm{non}-\mathrm{singular}\mathrm{matrix}.\mathrm{Therefore},\mathrm{it}\mathrm{is}\mathrm{invertible}. \\ \mathrm{Let}{C}_{ij}\mathrm{be}\mathrm{a}\mathrm{cofactor}\mathrm{of}{a}_{ij}\mathrm{in}A. \\ \mathrm{The}\mathrm{cofactors}\mathrm{of}\mathrm{element}\mathit{}A\mathrm{are}\mathrm{given}\mathrm{by} \\ {C}_{11}=0 \\ {C}_{12}=-2 \\ {C}_{21}=3 \\ {C}_{22}=1 \\ \therefore \mathrm{adj}A={\left[\begin{array}{cc}0& -2\\ 3& 1\end{array}\right]}^T=\left[\begin{array}{cc}0& 3\\ -2& 1\end{array}\right] \end{gathered}$$