If `A=[(3, 1),(-1, 2)]` , show that `A^2-5A+7I=O` . Use this to find `A^4`

Explanation

To solve the problem, we need to show that \( A^2 - 5A + 7I = O \) where \( A = \begin{pmatrix} 3 & 1 \\ -1 & 2 \end{pmatrix} \) and \( I \) is the identity matrix of the same order. Then we will use this result to find \( A^4 \).

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

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

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