Question

If \( A = \begin{bmatrix} i & 0 \\ 0 & -i \end{bmatrix}, B = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}, C = \begin{bmatrix} 0 & i \\ -i & 0 \end{bmatrix} \), then the expression \( A^2 + B^2 + C^2 \) is:

• A. \( A^2 + B^2 + C^2 = 3A^2B^2C^2 \)
• B. \( A^2 + B^2 + C^2 = 3ABC \)
• C. \( A^2 + B^2 + C^2 = 3I \)
• D. \( A^2 + B^2 + C^2 = 2ABC \)

Hint:

When squaring matrices, remember to perform matrix multiplication carefully and observe the resulting patterns.

Answer 1

The Correct Option is C

We are given the matrices \( A \), \( B \), and \( C \), and we need to calculate the expression \( A^2 + B^2 + C^2 \).
Step 1: Calculate \( A^2 \), \( B^2 \), and \( C^2 \). For matrix \( A \): \[ A = \begin{bmatrix} i & 0 \\ 0 & -i \end{bmatrix}, \quad A^2 = \begin{bmatrix} i & 0 \\ 0 & -i \end{bmatrix} \times \begin{bmatrix} i & 0 \\ 0 & -i \end{bmatrix} = \begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix} = -I \] For matrix \( B \): \[ B = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}, \quad B^2 = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix} \times \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix} = \begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix} = -I \] For matrix \( C \): \[ C = \begin{bmatrix} 0 & i \\ -i & 0 \end{bmatrix}, \quad C^2 = \begin{bmatrix} 0 & i \\ -i & 0 \end{bmatrix} \times \begin{bmatrix} 0 & i \\ -i & 0 \end{bmatrix} = \begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix} = -I \]
Step 2: Add \( A^2 \), \( B^2 \), and \( C^2 \). \[ A^2 + B^2 + C^2 = -I + (-I) + (-I) = -3I \] Thus, the expression \( A^2 + B^2 + C^2 \) equals \( -3I \), which is option (3), as \( I \) represents the identity matrix.