The problem asks for the standard matrix of the linear transformation $T: \mathbb{R}^2 \to \mathbb{R}^2$, which first reflects points through the horizontal $x_1$-axis and then reflects points through the line $x_1 = x_2$.

Step-by-step process:

1. Reflection through the $x_1$-axis:
   • The reflection matrix for the $x_1$-axis is:
   1 & 0 \\ 0 & -1 \end{pmatrix}$$ This reflects any point $$(x_1, x_2)$$ to $$(x_1, -x_2)$$.
2. Reflection through the line $x_1 = x_2$:
   • The reflection matrix for the line $x_1 = x_2$ swaps the coordinates $(x_1, x_2)$ to $(x_2, x_1)$. The matrix is:
   0 & 1 \\ 1 & 0 \end{pmatrix}$$
3. Combination of transformations: Since the transformation $T$ first reflects through the $x_1$-axis and then through the line $x_1 = x_2$, the standard matrix of $T$ is the product of the two matrices: $T = R_{x_1 = x_2} \cdot R_{x_1}$ Let's calculate this product: 0 & 1 \\ 1 & 0 \end{pmatrix} \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} = \begin{pmatrix} 0 \cdot 1 + 1 \cdot 0 & 0 \cdot 0 + 1 \cdot (-1) \\ 1 \cdot 1 + 0 \cdot 0 & 1 \cdot 0 + 0 \cdot (-1) \end{pmatrix} = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$$

Thus, the standard matrix of $T$ is: