\( \definecolor{colordef}{RGB}{249,49,84} \definecolor{colorprop}{RGB}{18,102,241} \)
Consider two transformations:
  • \(T_1\): A shear parallel to the x-axis that maps \((1,0)\) to \((1,0)\) and \((0,1)\) to \((2,1)\).
  • \(T_2\): A rotation of \(90^\circ\) anti-clockwise about the origin.
  1. Find the matrix \(\mathbf{A}\) representing \(T_1\).
  2. Find the matrix \(\mathbf{B}\) representing \(T_2\).
  3. Find the matrix \(\mathbf{C}\) representing the composite transformation \(T_1\) followed by \(T_2\).
  4. Find the image of the point \(P(1, 1)\) under this composite transformation.
  5. Find the coordinates of point \(S\) such that its image under this composite transformation is \(S'(-1, 5)\).
    (You are given that \(\mathbf{C}^{-1} = \begin{pmatrix} 2 & 1 \\ -1 & 0 \end{pmatrix}\)).

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