Show that the matrix `A=[[1, 2 ,2],[ 2 ,1, 2],[ 2, 2, 1]]` satisfies the equation `A^2-4A-5I_3=0` and hence find `A^(-1)`

Explanation

To show that the matrix \( A = \begin{bmatrix} 1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1 \end{bmatrix} \) satisfies the equation \( A^2 - 4A - 5I_3 = 0 \) and to find \( A^{-1} \), we will follow these steps:

### Step 1: Calculate \( A^2 \)

To find \( A^2 \), we multiply matrix \( A \) by itself:

\[
A^2 = A \cdot A = \begin{bmatrix} 1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1 \end{bmatrix} \cdot \begin{bmatrix} 1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1 \end{bmatrix}
...