Show that the matrix, `A=[[1, 0,-2],[-2,-1, 2],[ 3, 4, 1]]` satisfies the equation, `A^3-A^2-3A-I_3=O` . Hence, find `A^(-1)` .

Explanation

To show that the matrix \( A = \begin{bmatrix} 1 & 0 & -2 \\ -2 & -1 & 2 \\ 3 & 4 & 1 \end{bmatrix} \) satisfies the equation \( A^3 - A^2 - 3A - I_3 = O \), we will perform the following steps:

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

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

\[
A^2 = A \cdot A = \begin{bmatrix} 1 & 0 & -2 \\ -2 & -1 & 2 \\ 3 & 4 & 1 \end{bmatrix} \cdot \begin{bmatrix} 1 & 0 & -2 \\ -2 & -1 & 2 \\ 3 & 4 & 1 \end{bmatrix}
...