<s>[INST] Prove the matrix satisfies the equation $A^2 -4A-5I=0$ How to prove that
$$
A=\begin{bmatrix}
  1 & 2 & 2 \\
  2 & 1 & 2 \\
  2 & 2 & 1
\end{bmatrix}
$$
satisfies the equation $A^2 -4A-5I=0$?
 [/INST] $$A^2 = \left(
\begin{array}{ccc}
 9 & 8 & 8 \\
 8 & 9 & 8 \\
 8 & 8 & 9 \\
\end{array}
\right)$$
$$-4A = \left(
\begin{array}{ccc}
 -4 & -8 & -8 \\
 -8 & -4 & -8 \\
 -8 & -8 & -4 \\
\end{array}
\right)$$
$$A^2  - 4A = \left(
\begin{array}{ccc}
 5 & 0 & 0 \\
 0 & 5 & 0 \\
 0 & 0 & 5 \\
\end{array}
\right)$$
This is just as easy as any other way for this problem.
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