Question

Find the inverse of the matrix $\left|\begin{array}{ccc}3& 1& -1\\ 2& -2& 0\\ 1& 2& -1\end{array}\right|$

Answer 1

Let $A=\left|\begin{array}{ccc}3& 1& -1\\ 2& -2& 0\\ 1& 2& -1\end{array}\right|$

Then $\left|A\right|=\left|\begin{array}{ccc}3& 1& -1\\ 2& -2& 0\\ 1& 2& -1\end{array}\right|$ $=3\left(2\right)-1(-2)-1\left(6\right)=2$

We know,

$A^{-1}=\frac{1}{|A|}adj\left(A\right)$

$adj\left(A\right)=\left|\begin{array}{ccc}2& 2& 6\\ -1& -2& -5\\ -2& -2& -8\end{array}\right|$

$\therefore A^{-1}=\left|\begin{array}{ccc}1& 1& 3\\ -1/2& -1& -5/2\\ -1& -1& -4\end{array}\right|$