Question

If $A=\left|\begin{array}{cc}1& -1\\ 2& -1\end{array}\right|$ then prove that $A^{-1}=A^3$

Answer 1

$A-1=\frac{adjA}{|A|}$..........$\left(1\right)$

$\therefore |A|=-1-(-2)=1$..........$\left(2\right)$

$adjA={\left[\begin{array}{cc}-1& -2\\ 1& 1\end{array}\right]}^T=\left[\begin{array}{cc}-1& 1\\ -2& 1\end{array}\right]$............$\left(3\right)$

eqn $\left(1\right),\left(2\right)$ and $\left(3\right)$

$A^{-1}=\left[\begin{array}{cc}-1& 1\\ -2& 1\end{array}\right]$........$\left(4\right)$

$\because A^2=A.A=\left[\begin{array}{cc}1& -1\\ 2& -1\end{array}\right]\left[\begin{array}{cc}1& -1\\ 2& -1\end{array}\right]=\left[\begin{array}{cc}-1& 0\\ 0& -1\end{array}\right]$

$\because A^3=A^2.A=\left[\begin{array}{cc}-1& 0\\ 0& -1\end{array}\right]\left[\begin{array}{cc}1& -1\\ 2& -1\end{array}\right]$

$A^3=\left[\begin{array}{cc}-1& 1\\ -2& 1\end{array}\right]$........$\left(5\right)$

By eqn $\left(4\right)$ and $\left(5\right)$

$A^{-1}=A^3$