Question

If \[ A = \begin{bmatrix} \cos\alpha & \sin\alpha \\ -\sin\alpha & \cos\alpha \end{bmatrix}, \] verify that $A'A = I$.

Hint:

Orthogonal matrices satisfy $A'A = I$. Rotation matrices are classic examples of orthogonal matrices.

Answer 1

Step 1: Write $A'$, the transpose of $A$. \[ A' = \begin{bmatrix} \cos\alpha & -\sin\alpha \\ \sin\alpha & \cos\alpha \end{bmatrix} \] Step 2: Multiply $A'A$. \[ A'A = \begin{bmatrix} \cos\alpha & -\sin\alpha \\ \sin\alpha & \cos\alpha \end{bmatrix} \begin{bmatrix} \cos\alpha & \sin\alpha \\ -\sin\alpha & \cos\alpha \end{bmatrix} \] \[ = \begin{bmatrix} \cos^2\alpha + \sin^2\alpha & \cos\alpha\sin\alpha - \sin\alpha\cos\alpha \\ \sin\alpha\cos\alpha - \cos\alpha\sin\alpha & \sin^2\alpha + \cos^2\alpha \end{bmatrix} \] \[ = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} = I. \]

Final Answer: \[ \boxed{A'A = I} \]