We are tasked with finding the standard matrix $A$ for the linear transformation $T$, which involves two consecutive reflections. The transformation first reflects points through the $x$-axis, and then reflects points through the line $y = -x$.

Step-by-step solution:

1. Reflection through the $x$-axis:

   The matrix for reflecting a point through the $x$-axis is: $R_x = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}$ This matrix leaves the $x$-coordinate unchanged but negates the $y$-coordinate.

2. Reflection through the line $y = -x$:

   The matrix for reflecting a point through the line $y = -x$ is: $R_{y=-x} = \begin{bmatrix} 0 & -1 \\ -1 & 0 \end{bmatrix}$ This swaps the $x$- and $y$-coordinates and changes their signs.

3. Composite transformation:

   Since the transformation $T$ consists of first reflecting through the $x$-axis and then through the line $y = -x$, the standard matrix for $T$ is the product of these two matrices: $A = R_{y=-x} \cdot R_x$ Let's calculate this:

   $A = \begin{bmatrix} 0 & -1 \\ -1 & 0 \end{bmatrix} \cdot \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}$

4. Matrix multiplication: $A = \begin{bmatrix} 0 \times 1 + (-1) \times 0 & 0 \times 0 + (-1) \times (-1) \\ (-1) \times 1 + 0 \times 0 & (-1) \times 0 + 0 \times (-1) \end{bmatrix}$ Simplifying each element: $A = \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix}$

Thus, the standard matrix $A$ for the transformation $T$ is: $A = \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix}$

Would you like any further details or explanations on this problem? Here are five related questions you could explore:

1. What is the effect of reflecting through the line $y = x$ instead of $y = -x$?
2. How do you find the matrix for a general reflection through any line?
3. Can a reflection be represented by a rotation matrix?
4. How does matrix multiplication order affect transformations?
5. How would the transformation change if reflected first through the line $y = -x$ and then through the $x$-axis?

Tip: When performing transformations, always pay attention to the order of operations—matrix multiplication is not commutative!