A reflection matrix is usually not considered to be a rotation matrix, since rotation matrices have $\det\mathbf M=1$ and a reflection matrix can have $\det\mathbf M=-1$. An example of a reflection matrix is

\begin{equation}

\mathbf M =

\begin{pmatrix}

-1 & 0 \\

0 & 1

\end{pmatrix},

\end{equation}

whose determinant is $-1$.

Interestingly, the product of two reflection matrices become a rotation matrix. Assume we have another reflection matrix, i.e.,

\begin{equation}

\mathbf N =

\begin{pmatrix}

1 & 0 \\

0 & -1

\end{pmatrix}

\end{equation}

Their product becomes

\begin{equation}

\mathbf M\mathbf N =

\begin{pmatrix}

-1 & 0 \\

0 & 1

\end{pmatrix}

\begin{pmatrix}

1 & 0 \\

0 & -1

\end{pmatrix}

=

\begin{pmatrix}

-1 & 0 \\

0 & -1

\end{pmatrix},

\end{equation}

and as can be seen $\det(\mathbf M\mathbf N)=1$.

Read more about matrices in Chapter 6 in our book.

# Is a reflection matrix a rotation matrix?

1 Answers

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