Question

Find the inverse of the given matrix
$\left[\begin{array}{ccc}2& 1& 3\\ 4& -1& 0\\ -7& 2& 1\end{array}\right]$

Answer 1

Let $A=\left[\begin{array}{ccc}2& 1& 3\\ 4& -1& 0\\ -7& 2& 1\end{array}\right]$
We have, $|A|=\left[\begin{array}{ccc}2& 1& 3\\ 4& -1& 0\\ -7& 2& 1\end{array}\right]$
$=2(-1-0)-1(4-0)+3(8-7)=-2-4+3=-3$
Cofactors of A are
$A_{11}=-1-0=-1,A_{12}=-(4-0)=-4,A_{13}=8-7=1A_{21}=-(1-6)=5,A_{22}=2+21=23,A_{23}=-(4+7)=-11A_{31}=0+3=3,A_{32}=-(0-12)=12,A_{33}=-2-4=-6$
$\therefore adj\left(A\right)=\left[\begin{array}{ccc}-1& -4& 1\\ 5& 23& -11\\ 3& 12& -6\end{array}\right]=\left[\begin{array}{ccc}-1& 5& 3\\ -4& 23& 12\\ 1& -11& -6\end{array}\right]$
Now, $A^{-1}=\frac{1}{|A|}\left(adj\right(A\left)\right)=\frac{1}{-3}\left[\begin{array}{ccc}-1& 5& 3\\ -4& 23& 12\\ 1& -11& -6\end{array}\right]=\left[\begin{array}{ccc}\frac{1}{3}& -\frac{5}{3}& -1\\ \frac{4}{3}& -\frac{23}{3}& -4\\ -\frac{1}{3}& \frac{11}{3}& 2\end{array}\right]$