If `A=[[1, 0 ,2 ],[0 ,2 ,1],[2, 0 ,3]]`, prove that `A^3-6A^2+7A+2I=0`

Explanation

To prove that \( A^3 - 6A^2 + 7A + 2I = 0 \) for the matrix \( A = \begin{bmatrix} 1 & 0 & 2 \\ 0 & 2 & 1 \\ 2 & 0 & 3 \end{bmatrix} \), 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 & 0 & 2 \\ 0 & 2 & 1 \\ 2 & 0 & 3 \end{bmatrix} \cdot \begin{bmatrix} 1 & 0 & 2 \\ 0 & 2 & 1 \\ 2 & 0 & 3 \end{bmatrix}
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