Question:

Given: Problem statement: If $A = \begin{bmatrix} 3 & 2 \\ 1 & 1 \end{bmatrix}$, find numbers $a$ and $b$ such that $A^2 + aA + bI = O$. Hence find $A^{-1}$.
Step 1: Set up matrix system / determinant equation:
Write coefficient matrix $A$ and compute determinant $|A|$. [1.0 Mark]
Step 2: Evaluate adjugate matrix / line formula:
Calculate cofactors $A_{ij}$ and transpose to get $\text{adj } A$. [1.0 Mark]
Step 3: Evaluate inverse / solve for variables:
Apply $X = A^{-1} B$ to obtain target values. [1.0 Mark]
Conclusion: System solved successfully.

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🎯 Official CBSE Marking Scheme:
Setting up matrix system and evaluating |A|: 1.0 Mark
Evaluating cofactors and adjugate matrix: 1.0 Mark
Evaluating inverse and final solution: 1.0 Mark