Question

Find the inverse of matrix
$\left[\begin{array}{ccc}14& 3& -2\\ 6& 8& -1\\ 0& 2& -7\end{array}\right].$

Answer 1

Co-factors of; elements of first row of A are;
$\left|\begin{array}{cc}8& -1\\ 2& -7\end{array}\right|=-54,=-\left|\begin{array}{cc}6& -1\\ 0& 7\end{array}\right|=42,\left|\begin{array}{cc}6& 8\\ 0& 2\end{array}\right|=12$
Similarly co-factors of 2nd row are 17, -98, -28 whereas co-factors of 3rd row are 13, 2 94.
Adj. A = Transpose of the matrix formed by replacing each element by its co-factors.
$\therefore Adj.A=\left[\begin{array}{ccc}-54& 17& 13\\ 42& -98& 2\\ 12& -28& 94\end{array}\right]$
Also $|A|=\left|\begin{array}{ccc}14& 3& -2\\ 6& 8& -1\\ 0& 2& -7\end{array}\right|$
$=14(-56+2)-6(-21+4)=-654$
$\therefore$ $A^{-1}$ =$\frac{1}{|A|}Adj.A=\frac{1}{654}\left[\begin{array}{ccc}-54& 17& 13\\ 42& -98& 2\\ 12& -28& 94\end{array}\right]$
You can easily verify at $A{A}^{-1}=A^{-1}A=I$.