The element in the third row and first column of the inverse of the matrix $\left[\begin{array}{ccc}1 & -3 &…
The element in the third row and first column of the inverse of the matrix $\left[\begin{array}{ccc}1 & -3 & 2 \\ -3 & 3 & -1 \\ 2 & -1 & 0\end{array}\right]$ is
$-3$
$x=4$
$3$
$2$
Solution
We have $A=\left[\begin{array}{ccc}-1 & -3 & 2 \\ -3 & 3 & -1 \\ 2 & -1 & 0\end{array}\right] \begin{aligned}|A| &\Rightarrow =\left|\begin{array}{ccc}-1 & -3 & 2 \\ -3 & 3 & -1 \\ 2 & -1 & 0\end{array}\right| &=-(-1)+3(2)+2(3-6)=1+6-6=1 \end{aligned}$
We have to find $a_{31}$ in $A^{-1}$. So we will find $A_{13}$ in $A$.
Cofactor of $2=(-1)^{1+3}\left|\begin{array}{cc}-3 & 3 \\ 2 & -1\end{array}\right|=3-6=-3$
Hence $a_{31}$ in $A^{-1}=\frac{-3}{|A|}=\frac{-3}{1}=-3$