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

Step 1: Understanding the Concept:
We can solve this by calculating $A^2$ directly or by using the Characteristic Equation of the matrix.
Step 2: Formula Application:
$A^2 = \begin{pmatrix} 3 & -2
4 & -2 \end{pmatrix} \begin{pmatrix} 3 & -2
4 & -2 \end{pmatrix} = \begin{pmatrix} 9 - 8 & -6 + 4
12 - 8 & -8 + 4 \end{pmatrix} = \begin{pmatrix} 1 & -2
4 & -4 \end{pmatrix}$
Step 3: Explanation:
Now calculate $A^2 + I$: $\begin{pmatrix} 1 & -2
4 & -4 \end{pmatrix} + \begin{pmatrix} 1 & 0
0 & 1 \end{pmatrix} = \begin{pmatrix} 2 & -2
4 & -3 \end{pmatrix}$ Check option (c) $A - I$: $\begin{pmatrix} 3 & -2
4 & -2 \end{pmatrix} - \begin{pmatrix} 1 & 0
0 & 1 \end{pmatrix} = \begin{pmatrix} 2 & -2
4 & -3 \end{pmatrix}$ They match!
Step 4: Final Answer:
The result is $A - I$.