Question

If $$ A = \begin{pmatrix} -5 & -8 & 0 \\3 & 5 & 0 \\1 & 2 & -1 \end{pmatrix} $$ then $ A^2 $ is:

• A. Idempotent
• B. Nilpotent
• C. Symmetric
• D. Involutory

Hint:

A matrix is involutory if, when squared, it gives the identity matrix. Always check the result of \( A^2 \) to confirm if a matrix is involutory.

Answer 1

The Correct Option is D

To determine the property of \( A^2 \), we compute \( A^2 \) by performing matrix multiplication: \[ A^2 = \begin{pmatrix} -5 & -8 & 0 3 & 5 & 0 1& 2 & -1 \end{pmatrix} \times \begin{pmatrix} -5 & -8 & 0 \\3 & 5 & 0 \\1 & 2 & -1 \end{pmatrix} \] After matrix multiplication, we find that: \[ A^2 = I \quad (\text{the identity matrix}) \] This means the matrix \( A \) is involutory, as a matrix is involutory if \( A^2 = I \). 
Thus, the correct answer is 4. Involutory.