Question

Find the adjoint of the matrix $A=\left[\begin{array}{ccc}-1& -2& -2\\ 2& 1& -2\\ 2& -2& 1\end{array}\right]$ and hence show that $A\left(\mathrm{adj}A\right)=\left|A\right|I_3$.

Answer 1

$$\begin{gathered} A=\left[\begin{array}{ccc}-1& -2& -2\\ 2& 1& -2\\ 2& -2& 1\end{array}\right] \\ \mathrm{Now},\mathrm{to}\mathrm{find}\mathrm{Adj}.A \\ \begin{array}{ccc}{A}_{11}={\left(-1\right)}^{1+1}\left(-3\right)=-3 \\ {A}_{12}={\left(-1\right)}^{1+2}\left(6\right)=-6 \\ {A}_{13}={\left(-1\right)}^{1+3}\left(-6\right)=-6& A_{21}={\left(-1\right)}^{2+1}\left(-6\right)=6 \\ {A}_{22}={\left(-1\right)}^{2+2}\left(3\right)=3 \\ {A}_{23}={\left(-1\right)}^{2+3}\left(6\right)=-6& A_{31}={\left(-1\right)}^{3+1}\left(6\right)=6 \\ {A}_{32}={\left(-1\right)}^{3+2}\left(6\right)=-6 \\ {A}_{33}={\left(-1\right)}^{3+3}\left(3\right)=3\end{array} \\ \mathrm{Therefore}, \\ \mathrm{Adj}.A=\left[\begin{array}{ccc}-3& 6& 6\\ -6& 3& -6\\ -6& -6& 3\end{array}\right] \\ \mathrm{Now}, \\ \left|A\right|=\left|\begin{array}{ccc}-1& -2& -2\\ 2& 1& -2\\ 2& -2& 1\end{array}\right| \\ =-1\left(1-4\right)-2\left(-2-4\right)+2\left(4+2\right) \\ =3+12+12 \\ =27 \\ \mathrm{To}\mathrm{show}:A\left(adjA\right)=\left|A\right|I_3 \\ \mathrm{LHS}= \\ \left[\begin{array}{ccc}-1& -2& -2\\ 2& 1& -2\\ 2& -2& 1\end{array}\right]\left[\begin{array}{ccc}-3& 6& 6\\ -6& 3& -6\\ -6& -6& 3\end{array}\right] \\ =\left[\begin{array}{ccc}3+12+12& -6-6+12& -6+12-6\\ -6-6+12& 12+3+12& 12-6-6\\ -6+12-6& 12-6-6& 12+12+3\end{array}\right] \\ =\left[\begin{array}{ccc}27& 0& 0\\ 0& 27& 0\\ 0& 0& 27\end{array}\right] \\ =27\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right] \\ =\left|A\right|I_3 \\ =\mathrm{RHS} \\ \mathrm{Hence},A\left(\mathrm{adj}A\right)=\left|A\right|I_3. \end{gathered}$$