# Matrix Transformation across multiple planes

Let $T_1$ be a reflection of $\Bbb{R}^3$ in the xy plane, $T_2$ is a reflection of $\Bbb{R}^3$ in the xz plane. What is the standard matrix of transformation $T_2T_1$?

Here's my thinking so far:

Since the standard matrix for reflections in xy is $$\begin{bmatrix}1&0&0\\0&1&0\\0&0&0\\\end{bmatrix}$$

Similarly, standard matrix for orthogonal projection in the xz plane is $$\begin{bmatrix}1&0&0\\0&0&0\\0&0&1\\\end{bmatrix}$$

I could multiply $$\begin{bmatrix}1&0&0\\0&0&0\\0&0&1\\\end{bmatrix}*\begin{bmatrix}1&0&0\\0&1&0\\0&0&0\\\end{bmatrix}$$

to yield $$\begin{bmatrix}1&0&0\\0&0&0\\0&0&0\\\end{bmatrix}$$

Could someone confirm for me if this is a valid approach? Cheers.

## 1 Answer

Reflection in the $xy$ plane has matrix $$\left[\begin{array}{ccc}1&0&0\\0&1&0\\0&0&-1\end{array}\right]$$
and similar for reflection in the $xz$ plane.
Your matrices, as you say once, are for projection, rather than reflection. If you reflect in the $xy$ plane, the $x$ and $y$ values stay the same, but the $z$-value becomes its negative.
Your idea of multplying the matrices is correct.