Matrices & Determinants Question - None

Question

If A^T + A = I, where A = $\begin{pmatrix} \cos\theta & \sin\theta \ -\sin\theta & \cos\theta \end{pmatrix}$, find the value of $\theta$.

Accepted Answer

Step 1: Compute A^T:

$\mathbf{A}^T = \begin{pmatrix} \cos heta & -\sin heta \ \sin heta & \cos heta \end{pmatrix}$

Step 2: Add A^T + A:

$\mathbf{A}^T + \mathbf{A} = \begin{pmatrix} \cos heta & -\sin heta \ \sin heta & \cos heta \end{pmatrix} + \begin{pmatrix} \cos heta & \sin heta \ -\sin heta & \cos heta \end{pmatrix} = \begin{pmatrix} 2\cos heta & 0 \ 0 & 2\cos heta \end{pmatrix}$

Step 3: Set equal to I:

$\begin{pmatrix} 2\cos heta & 0 \ 0 & 2\cos heta \end{pmatrix} = \begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix}$

Step 4: Solve for $\cos heta$:

$2\cos heta = 1 \Rightarrow \cos heta = \frac{1}{2} = \cos\frac{\pi}{3}$

∴ $ heta = 2n\pi \pm \frac{\pi}{3}, \quad n \in \mathbb{I}$

Question:

Step-by-Step Solution

Master Matrices & Determinants with Mathbee