Examples of Linear Transformation problem Assume that T is a linear transformation. Find the standard matrix of T. 
 $T:\mathbb{R}^2 \rightarrow\mathbb{R}^2$
first reflects points through the line $x_2$=$x_1$ and then reflects points through the horizontal $x_1$-axis.
My Solution , that is incorrect :-
The standard matrix for the reflection through the line $x_2$=$x_1$ is 
\begin{bmatrix}0&1\\1&0\end{bmatrix}
The standard matrix for the reflection through the horizontal $x_1$-axis is :
\begin{bmatrix}1&0\\0&-1\end{bmatrix}
When we multiply this we get :
\begin{bmatrix}0&-1\\1&0\end{bmatrix}
This answer is being rejected. Can you please advise me what am I doing wrong?
Thank you.
 A: The task is simple . . .

Let $B$ be the matrix for the transformation which reflects a point about the line $x_2=x_1$. Then
$$B = \begin{bmatrix}0&1\\1&0\end{bmatrix}$$

Let $A$ be the matrix for the transformation which reflects a point about the $x_1$-axis. Then
$$A = \begin{bmatrix}1&0\\0&-1\end{bmatrix}$$
The matrix for the transformation which first reflects a point about the line $x_2=x_1$, and then reflects the result about the $x_1$-axis is just
$$AB = \begin{bmatrix}1&0\\0&-1\end{bmatrix}\begin{bmatrix}0&1\\1&0\end{bmatrix}=\begin{bmatrix}0&1\\-1&0\end{bmatrix}$$

To test it, let $v={\displaystyle{\begin{bmatrix}x_1\\x_2\end{bmatrix}}}$.

First apply the transformations by hand . . .


*

*Start with $$\begin{bmatrix}x_1\\x_2\end{bmatrix}$$

*Next, reflect about the line $x1=x2$:
$$\begin{bmatrix}x_1\\x_2\end{bmatrix} \rightarrow \begin{bmatrix}x_2\\x_1\end{bmatrix}$$

*Next, reflect the above result about the $x_1$-axis:
$$\begin{bmatrix}x_2\\x_1\end{bmatrix} \rightarrow \begin{bmatrix}x_2\\-x_1\end{bmatrix}$$


Next, check to see if $(AB)v$ yields the same result:
$$(AB)v = \begin{bmatrix}0&1\\-1&0\end{bmatrix} \begin{bmatrix}x_1\\x_2\end{bmatrix} = \begin{bmatrix}x_2\\-x_1\end{bmatrix}$$
so it checks.
