Question

If \( A = \begin{pmatrix} 0 & 0 & -1 \\ 0 & -1 & 0 \\ -1 & 0 & 0 \end{pmatrix} \), find \( I + A^2 \), where \( I \) is the identity matrix.

Hint:

For matrix operations like \( A^2 \), carry out the matrix multiplication step by step. The identity matrix \( I \) is often used to simplify expressions in linear algebra.

Answer 1

Step 1: Begin by calculating \( A^2 \). Multiply the matrix \( A \) by itself: \[ A = \begin{pmatrix} 0 & 0 & -1 \\ 0 & -1 & 0 \\ -1 & 0 & 0 \end{pmatrix}. \] Perform the matrix multiplication \( A \times A \): \[ A^2 = \begin{pmatrix} 0 & 0 & -1 \\ 0 & -1 & 0 \\ -1 & 0 & 0 \end{pmatrix} \times \begin{pmatrix} 0 & 0 & -1 \\ 0 & -1 & 0 \\ -1 & 0 & 0 \end{pmatrix}. \] Step 2: Carry out the multiplication: \[ A^2 = \begin{pmatrix} (0)(0) + (0)(0) + (-1)(-1) & (0)(0) + (0)(-1) + (-1)(0) & (0)(-1) + (0)(0) + (-1)(0) \\ (0)(0) + (-1)(0) + (0)(-1) & (0)(0) + (-1)(-1) + (0)(0) & (0)(-1) + (-1)(0) + (0)(0) \\ (-1)(0) + (0)(0) + (0)(-1) & (-1)(0) + (0)(-1) + (0)(0) & (-1)(-1) + (0)(0) + (0)(0) \end{pmatrix} \] Simplifying: \[ A^2 = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}. \] Step 3: Now, compute \( I + A^2 \), where \( I \) is the identity matrix: \[ I = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}, \quad A^2 = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}. \] Thus: \[ I + A^2 = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} + \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} = \begin{pmatrix} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 2 \end{pmatrix}. \] This is equal to \( 2I \).