If `A=[(1, 2 ,2 ),(2 ,1 ,2),( 2 ,2, 1)]` , then prove that `A^2-4A-5I=O` .

Explanation

To prove that \( A^2 - 4A - 5I = O \), where \( A = \begin{pmatrix} 1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1 \end{pmatrix} \) and \( I \) is the identity matrix, we will follow these steps:

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

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

\[
A^2 = A \times A = \begin{pmatrix} 1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1 \end{pmatrix} \times \begin{pmatrix} 1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1 \end{pmatrix}
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