Question

If \( A = \begin{bmatrix} a & b \\ c & -a \end{bmatrix} \) is such that \( A^2 = I \), then _____

• A. \( 1 + a^2 + bc = 0 \)
• B. \( 1 - a^2 - bc = 0 \)
• C. \( 1 - a^2 + bc = 0 \)
• D. \( 1 + a^2 - bc = 0 \)

Hint:

For \( A^2 = I \), diagonal elements must be 1 and off-diagonal must be 0.

Answer 1

The Correct Option is B

Concept: Compute \( A^2 \) and equate with identity matrix.
Step 1: Find \( A^2 \). \[ A^2 = \begin{bmatrix} a & b \\ c & -a \end{bmatrix} \begin{bmatrix} a & b \\ c & -a \end{bmatrix} = \begin{bmatrix} a^2 + bc & 0 \\ 0 & a^2 + bc \end{bmatrix} \]
Step 2: Equate with identity. \[ A^2 = I \Rightarrow a^2 + bc = 1 \]
Step 3: \[ 1 - a^2 - bc = 0 \]