Question

If $A=\begin{array}{ccc}1& 3& 3\\ 1& 4& 3\\ 1& 3& 4\end{array}$, then verify A. adj A = |A|.I Also find $A^{-1}$

Answer 1

If $A=\left|\begin{array}{ccc}1& 3& 3\\ 1& 4& 3\\ 1& 3& 4\end{array}\right||A|=1(12+12)-(12-9)+(9-12)$

$=-3-3=-6$

$A.\left(Ady\right)=|A|$

$A^{-1}=\frac{1}{|A|}$ adjA verify

$A^{-1}=\frac{1}{-6}\left|\begin{array}{ccc}12-12& 9-12& 9-12\\ 3-4& 4-3& 3-3\\ 3-4& 3-3& 4-3\end{array}\right|$

$=\frac{-1}{6}\left|\begin{array}{ccc}0& -3& -3\\ -1& 1& 0\\ -1& 0& 1\end{array}\right|$

$=\frac{1}{6}\left|\begin{array}{ccc}0& 3& 3\\ 1& -1& 0\\ 1& 0& -1\end{array}\right|$