Matrix for a reflection in y=x followed by a vertical stretch

Key takeaway: For composite transformations, order matters. First identify the matrix for the reflection, then the matrix for the stretch, and multiply them in the correct order so the second transformation acts on the result of the first.

A reflection in the line $y=x$ sends

$$\begin{pmatrix}x\\y\end{pmatrix} \mapsto \begin{pmatrix}y\\x\end{pmatrix},$$

so its matrix is

$$R=\begin{pmatrix}0&1\\1&0\end{pmatrix}.$$

A stretch by factor 3 in the positive $y$-direction sends

$$\begin{pmatrix}x\\y\end{pmatrix} \mapsto \begin{pmatrix}x\\3y\end{pmatrix},$$

so its matrix is

$$S=\begin{pmatrix}1&0\\0&3\end{pmatrix}.$$

Because the reflection happens first and the stretch happens second, the combined matrix is

$$SR=\begin{pmatrix}1&0\\0&3\end{pmatrix} \begin{pmatrix}0&1\\1&0\end{pmatrix} = \begin{pmatrix}0&1\\3&0\end{pmatrix}.$$

So the required transformation matrix is

$$\boxed{\begin{pmatrix}0&1\\3&0\end{pmatrix}}.$$

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Pitfalls the pros know 👇 The biggest trap is reversing the order of multiplication. If you multiply the matrices the other way around, you describe a different transformation. Also, the vertical stretch only changes the $y$-coordinate, not both coordinates.

What if the problem changes? If the stretch were instead in the direction of the positive $x$-axis by factor 3, the stretch matrix would be

$$\begin{pmatrix}3&0\\0&1\end{pmatrix},$$

and the combined matrix would become

$$\begin{pmatrix}3&0\\0&1\end{pmatrix} \begin{pmatrix}0&1\\1&0\end{pmatrix} = \begin{pmatrix}0&3\\1&0\end{pmatrix}.$$

Tags: linear transformation, reflection matrix, matrix multiplication order