Question

If \( A = \begin{bmatrix} 3 & -2 \\ 4 & -2 \end{bmatrix} \), then \( A^2 + I = \) _____

• A. \( A - 2I \)
• B. \( A + I \)
• C. \( A - I \)
• D. \( I - A \)

Hint:

Always verify options by direct substitution in matrix problems.

Answer 1

The Correct Option is C

Concept: Direct matrix multiplication.
Step 1: Compute \( A^2 \). \[ A^2 = \begin{bmatrix} 3 & -2 \\ 4 & -2 \end{bmatrix} \begin{bmatrix} 3 & -2 \\ 4 & -2 \end{bmatrix} = \begin{bmatrix} 1 & -2 \\ 4 & -4 \end{bmatrix} \]
Step 2: Add identity matrix. \[ A^2 + I = \begin{bmatrix} 1 & -2 \\ 4 & -4 \end{bmatrix} + \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} = \begin{bmatrix} 2 & -2 \\ 4 & -3 \end{bmatrix} \]
Step 3: \[ A - I = \begin{bmatrix} 3 & -2 \\ 4 & -2 \end{bmatrix} - \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} = \begin{bmatrix} 2 & -2 \\ 4 & -3 \end{bmatrix} \] \[ \Rightarrow A^2 + I = A - I \]