For each of the following matrices, find its determinant to see whether it produces a rotation or a reflection. If a rotation, find the axis and angle of rotation. If a reflection, find the reflecting plane and the rotation (if any) about the normal to this plane.

_{$\mathit{M}\mathbf{=}\mathbf{\left(}\begin{array}{ccc}\mathbf{0}& \mathbf{-}\mathbf{1}& \mathbf{0}\\ \mathbf{1}& \mathbf{0}& \mathbf{0}\\ \mathbf{0}& \mathbf{0}& \mathbf{-}\mathbf{1}\end{array}\mathbf{\right)}$}

If a determinant whose absolute value is unity, then Rotation matrices have a determinant of +1, and reflection matriceshave a determinant of −1. The set of all orthogonal two-dimensional matrices together with matrix multiplication form the orthogonal group.

Reflection rules: Reflection in the y-axis$\left(\begin{array}{cc}-1& 0\\ 0& 1\end{array}\right)$,Reflection in the x-axis$\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right)$,

Reflection in the line$y=x\mathbf{\left(}\begin{array}{cc}\mathbf{0}& \mathbf{-}\mathbf{1}\\ \mathbf{-}\mathbf{1}& \mathbf{0}\end{array}\mathbf{\right)}$, Reflection in$y=x\mathbf{\left(}\begin{array}{cc}\mathbf{0}& \mathbf{-}\mathbf{1}\\ \mathbf{-}\mathbf{1}& \mathbf{0}\end{array}\mathbf{\right)}$.

The given matrices are$M=\left(\begin{array}{ccc}0& -1& 0\\ 1& 0& 0\\ 0& 0& -1\end{array}\right)$ and$r=\left(\begin{array}{c}x\\ y\\ z\end{array}\right)$.

The determinant of matrices is equal to , the reflection plane needs to be determined.

To solve these equations for the vector perpendicular to the plane of reflection the equation will become $Mr=-r.$.

localid="1658995440016" $M=\left(\begin{array}{ccc}0& -1& 0\\ 1& 0& 0\\ 0& 0& -1\end{array}\right)\left(\begin{array}{c}x\\ y\\ z\end{array}\right)=-\left(\begin{array}{c}x\\ y\\ z\end{array}\right)$

Multiply the matrices.

$\begin{array}{c}-x=-x\\ y=-y\\ -z=-z\end{array}$

From the above definition, it shows reflection in the x-y plane. It also means that the vector perpendicular to the plane of reflection is k. Therefore, the matrix determinant is -1 which produces reflection in the x-y plane.