Question

Find the inverse of $A=\left[\begin{array}{cc}1& -1\\ 2& 3\end{array}\right]$ by using Adjoint method.

Answer 1

$\begin{array}{c}A=\left[\begin{array}{cc}1& -1\\ 2& 3\end{array}\right]\\ A^{-1}=\frac{1}{\left|A\right|}adjA\to \left(i\right)\\ \left|A\right|=3+2=5\to \left(ii\right)\\ \begin{array}{c}{A}_{11}=3\\ A_{12}=-2\end{array}|\begin{array}{c}{A}_{21}=1\\ A_{22}=1\end{array}\\ adjA=\left[\begin{array}{cc}3& 1\\ -2& 1\end{array}\right]\to \left(iii\right)\\ from\left(i\right),\left(ii\right)and\left(iii\right)\\ A^{-1}=\frac{1}{5}\left[\begin{array}{cc}3& 1\\ -2& 1\end{array}\right]\end{array}$