If $\mathrm{A}=\left[\begin{array}{ccc}1& 2& 2\\ 2& 1& 2\\ 2& 2& 1\end{array}\right] \mathrm{then} {\mathrm{A}}^3-4{\mathrm{A}}^2-6\mathrm{A}=$

$$\begin{gathered} {\mathrm{A}}^2=\left[\begin{array}{ccc}1& 2& 2\\ 2& 1& 2\\ 2& 2& 1\end{array}\right] \left[\begin{array}{ccc}1& 2& 2\\ 2& 1& 2\\ 2& 2& 1\end{array}\right] \\ =\left[\begin{array}{ccc}9& 8& 8\\ 8& 9& 8\\ 8& 8& 9\end{array}\right] \\ \therefore {\mathrm{A}}^3-4{\mathrm{A}}^2-6\mathrm{A}=\left[\begin{array}{ccc}41& 42& 42\\ 42& 41& 42\\ 42& 42& 41\end{array}\right]-\left[\begin{array}{ccc}36& 32& 32\\ 32& 36& 32\\ 32& 32& 36\end{array}\right]-\left[\begin{array}{ccc}6& 12& 12\\ 12& 6& 12\\ 12& 12& 6\end{array}\right] \end{gathered}$$

$$\begin{gathered} =\left[\begin{array}{ccc}-1& -2& -2\\ -2& -1& -2\\ -2& -2& -1\end{array}\right] \\ =-\left[\begin{array}{ccc}1& 2& 2\\ 2& 1& 2\\ 2& 2& 1\end{array}\right] \\ =-\mathrm{A} \end{gathered}$$