Question

Find the inverse of $A=\left[\begin{array}{ccc}0& 1& 2\\ 1& 2& 3\\ 3& 1& 1\end{array}\right]$ if
$A^{-1}=\left[\begin{array}{ccc}1/2& -1/2& 1/2\\ a& 3& b\\ c& -3/2& 1/2\end{array}\right]$
Find $|abc|?$

Answer 1

Given $A=\left[\begin{array}{ccc}0& 1& 2\\ 1& 2& 3\\ 3& 1& 1\end{array}\right]$
$\therefore |A|=0.(2-3)-1(1-9)+2(1+6)$
$=0+8-10$
$=-2\ne 0$
If $C$ be the matrix of cofactors of the elements in $|A|$
$\therefore C=\left[\begin{array}{ccc}{C}_{11}& C_{12}& C_{13}\\ C_{21}& C_{22}& C_{23}\\ C_{31}& C_{32}& C_{33}\end{array}\right]$
$C_{11}=\left|\begin{array}{cc}2& 3\\ 1& 1\end{array}\right|=-1;C_{12}=-\left|\begin{array}{cc}1& 3\\ 3& 1\end{array}\right|=8;C_{13}=\left|\begin{array}{cc}1& 2\\ 3& 1\end{array}\right|=-5$
$C_{21}=-\left|\begin{array}{cc}1& 2\\ 1& 1\end{array}\right|=1;C_{22}=\left|\begin{array}{cc}0& 2\\ 3& 1\end{array}\right|=-6;C_{23}=-\left|\begin{array}{cc}0& 1\\ 3& 1\end{array}\right|=3$
$C_{31}=\left|\begin{array}{cc}1& 2\\ 2& 3\end{array}\right|=-1;C_{32}=-\left|\begin{array}{cc}0& 2\\ 1& 3\end{array}\right|=2;C_{33}=\left|\begin{array}{cc}0& 1\\ 1& 2\end{array}\right|=-1$
$\therefore C=\left[\begin{array}{ccc}-1& 8& -5\\ 1& -6& 3\\ -1& 2& -1\end{array}\right]$
$\therefore Adj\left(A\right)=C^{\prime }=\left[\begin{array}{ccc}-1& 1& -1\\ 8& -6& 2\\ -5& 3& -1\end{array}\right]$
Hence, $A^{-1}=\frac{AdjA}{|A|}=-\frac{1}{2}\left[\begin{array}{ccc}-1& 1& -1\\ 8& -6& -2\\ -5& 3& -1\end{array}\right]$
$=\left[\begin{array}{ccc}1/2& -1/2& 1/2\\ -4& 3& -1\\ 5/2& -3/2& 1/2\end{array}\right]$

$\Rightarrow |abc|=10$