The image contains three multiple-choice questions related to matrices:

Question 1

Which matrix is not orthogonal?

To determine if a matrix is orthogonal, it must satisfy $A^T A = I$, where $A^T$ is the transpose of $A$, and $I$ is the identity matrix.

Let's analyze each option for orthogonality:

• (a) $\begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}$
• (b) $\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$
• (c) $\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$
• (d) $\begin{pmatrix} 1 & 0 \\ 1 & 0 \end{pmatrix}$

Only option (d) does not satisfy $A^T A = I$. Answer: (d)

Question 2

Which matrix rotates a three-by-one column vector an angle $\theta$ counterclockwise around the Z-axis?

For a rotation around the Z-axis, the rotation matrix is: