For the matrix A=`[[3,2],[1,1]]`, find the numbers a and b such that `A^2+aA+bI=O`.

Explanation

To solve the problem, we need to find the values of \( a \) and \( b \) such that the equation \( A^2 + aA + bI = O \) holds true for the matrix \( A = \begin{pmatrix} 3 & 2 \\ 1 & 1 \end{pmatrix} \).

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

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

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