Question

For the matrix $A=\left[\begin{array}{ccc}1& -1& 1\\ 2& 3& 0\\ 18& 2& 10\end{array}\right]$, show that A (adj A) = O.

Answer 1

$$\begin{gathered} A=\left[\begin{array}{ccc}1& -1& 1\\ 2& 3& 0\\ 18& 2& 10\end{array}\right] \\ \mathrm{Now}, \\ {C}_{11}=\left|\begin{array}{cc}3& 0\\ 2& 10\end{array}\right|=30C_{12}=-\left|\begin{array}{cc}2& 0\\ 18& 10\end{array}\right|=-20C_{13}=\left|\begin{array}{cc}2& 3\\ 18& 2\end{array}\right|=-50 \\ {C}_{21}=-\left|\begin{array}{cc}-1& 1\\ 2& 10\end{array}\right|=12C_{22}=\left|\begin{array}{cc}1& 1\\ 18& 10\end{array}\right|=-8C_{23}=-\left|\begin{array}{cc}1& -1\\ 18& 2\end{array}\right|=-20 \\ {C}_{31}=\left|\begin{array}{cc}-1& 1\\ 3& 0\end{array}\right|=-3C_{32}=-\left|\begin{array}{cc}1& 1\\ 2& 0\end{array}\right|=2C_{33}=\left|\begin{array}{cc}1& -1\\ 2& 3\end{array}\right|=5 \\ \mathrm{adj}A={\left[\begin{array}{ccc}30& -20& -50\\ 12& -8& -20\\ -3& 2& 5\end{array}\right]}^T=\left[\begin{array}{ccc}30& 12& -3\\ -20& -8& 2\\ -50& -20& 5\end{array}\right] \\ \therefore A(\mathrm{adj}A)=\left[\begin{array}{ccc}1& -1& 1\\ 2& 3& 0\\ 18& 2& 10\end{array}\right]\left[\begin{array}{ccc}30& 12& -3\\ -20& -8& 2\\ -50& -20& 5\end{array}\right]=\left[\begin{array}{ccc}30+20-50& 12+18-20& -3-2+5\\ 60-60-0& 24-24-0& -6+6+0\\ 540-40-500& 216-16-200& -54+4+50\end{array}\right]=\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right] \end{gathered}$$