If `A=[[2,-1, 1],[-1 ,2,-1],[ 1, -1, 2]]`.Verify that `A^3-6A^2+9A-4I=0`and hence find `A^(-1)`.

Explanation

To verify that \( A^3 - 6A^2 + 9A - 4I = 0 \) for the matrix \( A = \begin{bmatrix} 2 & -1 & 1 \\ -1 & 2 & -1 \\ 1 & -1 & 2 \end{bmatrix} \), 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} 2 & -1 & 1 \\ -1 & 2 & -1 \\ 1 & -1 & 2 \end{bmatrix} \cdot \begin{bmatrix} 2 & -1 & 1 \\ -1 & 2 & -1 \\ 1 & -1 & 2 \end{bmatrix}
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