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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? 

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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.

Matrix S represents a reflection on line y = x, matrix T represents a rotation through positive 90° centered at (0,0). A triangle whose vertices are A(0,0), B(1,-2), and C(0,1) is subjected to these transformations such that the triangle A'B'C' is the image of ABC under transformation matrix S and that A''B''C'' is the image of A'B'C' under transformation matrix T.

(a) Plot the three triangles.
(b) Find a single matrix that will map A''B''C'' onto ABC.
(c) Find a single matrix that will map A'B'C' onto ABC.
(d) If triangle ABC is sheared with shear factor 2 with y-axis invariant, find coordinates of the image.

Explore the effects of reflection, rotation, and shear transformations on a triangle with vertices at A(0,0), B(1,-2), and C(0,1). Learn how to apply matrix transformations, plot resulting triangles, and find composite matrices for mapping transformations. Ideal for advanced high school or college-level linear algebra.

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