Answer
(3,1)- element in the inverse of the given matrix is $-\frac{5}{7}$
Work Step by Step
$ \det(A)=\begin{bmatrix}
2 & 0 & -1\\
2 & 1 & 1 \\
3 & -1 & 0
\end{bmatrix}=(-1).(-5)+(-2).(-1)+2.0=7$
To find (3,1)- element in the inverse of the given matrix A:
$(A^{-1})_{31}=\frac{1}{\det(A)}adj.A_{31}=\frac{1}{\det(A)}.C_{31}$
Now find the cofactor $C_{11}$ of the matrix A:
$C_{31}=(-1)^{3+1}.M_{32}=1.\begin{vmatrix}
2 & 1\\
3& -1
\end{vmatrix}=(-5).1=-5$
Hence, $(A^{-1})_{31}=\frac{1}{\det(A)}.C_{31}=\frac{1}{7}.(-5)=-\frac{5}{7}$
(3,1)- element in the inverse of the given matrix is $-\frac{5}{7}$
