Prove \(A^2 = B^2 = C^2 = I_2\)
Question:
Given \[ A=\begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}, \quad B=\begin{bmatrix}1 & 0 \\ 0 & -1\end{bmatrix}, \quad C=\begin{bmatrix}0 & 1 \\ 1 & 0\end{bmatrix} \] show that: \[ A^2 = B^2 = C^2 = I_2 \]
Given \[ A=\begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}, \quad B=\begin{bmatrix}1 & 0 \\ 0 & -1\end{bmatrix}, \quad C=\begin{bmatrix}0 & 1 \\ 1 & 0\end{bmatrix} \] show that: \[ A^2 = B^2 = C^2 = I_2 \]
Solution:
Step 1: Compute \(A^2\)
\[ A^2 = \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix} \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix} = \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix} = I_2 \]Step 2: Compute \(B^2\)
\[ B^2 = \begin{bmatrix}1 & 0 \\ 0 & -1\end{bmatrix} \begin{bmatrix}1 & 0 \\ 0 & -1\end{bmatrix} = \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix} = I_2 \]Step 3: Compute \(C^2\)
\[ C^2 = \begin{bmatrix}0 & 1 \\ 1 & 0\end{bmatrix} \begin{bmatrix}0 & 1 \\ 1 & 0\end{bmatrix} = \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix} = I_2 \]Conclusion:
\[ A^2 = B^2 = C^2 = I_2 \]Hence proved.