This problem involves multiple geometric transformations on a triangle with initial vertices $A(0,0)$, $B(1,-2)$, and $C(0,1)$. Here’s a breakdown of the tasks:

1. Task (a): Plot the three triangles.

   • Start by plotting the initial triangle $ABC$ based on the provided vertices.
   • Apply the transformation matrix $S$ for reflection over $y = x$ and matrix $T$ for a 90° rotation about the origin to find the new vertices for triangles $A'B'C'$ and $A''B''C''$.

2. Task (b): Find a single matrix that maps $A''B''C''$ onto $ABC$.

   • You need to combine the inverse transformations of $S$ and $T$ to bring $A''B''C''$ back to $ABC$.

3. Task (c): Find a single matrix that maps $A'B'C'$ onto $ABC$.

   • Similar to (b), find the inverse of the transformation that took $ABC$ to $A'B'C'$.

4. Task (d): Shear transformation on triangle $ABC$ with shear factor 2 and the $y$-axis invariant.

   • This involves applying a shear matrix where the $x$-coordinates change but the $y$-coordinates remain the same.

Would you like detailed solutions for each part, or guidance on specific sections?

Here are some related questions to deepen your understanding:

1. What is the effect of a reflection transformation over different lines, such as $y = x$ vs. $x = 0$?
2. How do we determine the inverse of a transformation matrix?
3. What are the implications of applying transformations in different sequences?
4. How does a shear transformation differ from a rotation or reflection?
5. Can the combination of transformations be represented by a single matrix, and how is it calculated?

Tip: When working with transformations, always check the order of multiplication for matrices, as matrix multiplication is not commutative.