Question

If $A=\left[\begin{array}{cc}2& 3\\ 5& -2\end{array}\right]$ be such that $A^{-1}=kA$, then find the value of $k$.

Answer 1

$A=\left[\begin{array}{c}23\\ 5-2\end{array}\right]$

Co-factors of A are:-

$A_{11}=(-1{)}^2(-2)=-2$

$A_{12}=(-1{)}^3(5)=-5$

$A_{21}=(-1{)}^3(3)=-3$

$A_{22}=(-1{)}^4(2)=2$

$AdjA={\left[\begin{array}{c}-2-5\\ -32\end{array}\right]}^T$

$=\left[\begin{array}{c}-2-3\\ -52\end{array}\right]$

$\mid A\mid =-19$

$A^{-1}=\left[\begin{array}{c}\frac{2}{19}\frac{5}{19}\\ \frac{3}{19}\frac{-2}{19}\end{array}\right]$

$=K\left[\begin{array}{c}25\\ 3-2\end{array}\right]$

$=\frac{1}{19}\left[\begin{array}{c}25\\ 3-2\end{array}\right]=K\left[\begin{array}{c}25\\ 3-2\end{array}\right]$

Comparing, we get $K=\frac{1}{19}$