Question

If $A=\left[\begin{array}{ccc}1& 3& 3\\ 1& 4& 3\\ 1& 3& 4\end{array}\right]$, then find $A^{-1}$.

Answer 1

Given $A=\left[\begin{array}{ccc}1& 3& 3\\ 1& 4& 3\\ 1& 3& 4\end{array}\right]$

The determinant of $A$ will be $1(16-9)-3(4-3)+3(3-4)=7-3-3=1$

Adjoint of $A$ is transpose of matrix $\left[\begin{array}{ccc}7& -1& -1\\ -3& 1& 0\\ -3& 0& 1\end{array}\right]$

Therefore adjoint of $A$ is $\left[\begin{array}{ccc}7& -3& -3\\ -1& 1& 0\\ -1& 0& 1\end{array}\right]$

Therefore inverse will be $\left[\begin{array}{ccc}7& -3& -3\\ -1& 1& 0\\ -1& 0& 1\end{array}\right]$