Understand and use composition of linear transformations and the corresponding matrix products.

A linear transformation \(T_1\) is represented by the matrix \(A = \begin{pmatrix} 2 & -1 \\ 0 & 3 \end{pmatrix}\), and a second linear transformation \(T_2\) is represented by the matrix \(B = \begin{pmatrix} 1 & 2 \\ -1 & 1 \end{pmatrix}\). a) Determine the matrix \(C\) that represents the composition \(T_2 \circ T_1\). b) Use your result from part (a) to determine the image of the point \(P(3, -2)\) under the composition \(T_2 \circ T_1\).

A game designer applies two transformations to position a sprite on screen. First, transformation \(T_1\) rotates points anticlockwise by \(90^\circ\) about the origin. Second, transformation \(T_2\) reflects points in the line \(y = x\). The initial position of a sprite corner is at point \(P(3, -2)\). Determine the final position of point \(P\) after both transformations are applied in the order \(T_1\) followed by \(T_2\), by using the composition of the corresponding transformation matrices. Show evidence of the matrix product in your solution.

Consider two linear transformations of the plane. Transformation \(T_1\) is a reflection in the \(x\)-axis, and transformation \(T_2\) is an anticlockwise rotation of \(90^\circ\) about the origin. Determine the single matrix that represents the composite transformation \(T_2(T_1(\mathbf{x}))\), where the transformation \(T_1\) is applied first, followed by \(T_2\). Show evidence of the matrix product used in your solution.