Question

Find the inverse of the matrix $\left[\begin{array}{cc}3& -2\\ -7& 5\end{array}\right].$

Answer 1

$$\begin{gathered} \left|A\right|=\left|\begin{array}{cc}3& -2\\ -7& 5\end{array}\right|=1\ne 0 \\ A\mathrm{is}\mathrm{a}\mathrm{non}-\mathrm{singular}\mathrm{matrix}.\mathrm{Therefore},\mathrm{it}\mathrm{is}\mathrm{invertible}. \\ \mathrm{Let}{C}_{ij}\mathrm{be}\mathrm{a}\mathrm{cofactor}\mathrm{of}{a}_{ij}inA. \\ \mathrm{The}\mathrm{cofactors}\mathrm{of}\mathrm{element}A\mathrm{are}\mathrm{given}\mathrm{by} \\ {C}_{11}=5 \\ {C}_{12}=7 \\ {C}_{21}=2 \\ {C}_{22}=3 \\ \therefore A^{-1}=\frac{1}{\left|A\right|}{\left[\begin{array}{cc}5& 7\\ 2& 3\end{array}\right]}^T=\left[\begin{array}{cc}5& 2\\ 7& 3\end{array}\right] \end{gathered}$$