Commutativity of transformations

8 Three transformations, \(T _ { A } , T _ { B }\) and \(T _ { C }\), are represented by the matrices \(A , B\) and \(\mathbf { C }\) respectively. You are given that \(\mathbf { A } = \left( \begin{array} { l l } 1 & 0 \\ 2 & 3 \end{array} \right)\) and \(\mathbf { B } = \left( \begin{array} { c c } 1 & 0 \\ 0 & - 1 \end{array} \right)\).

1. Find the matrix which represents the inverse transformation of \(T _ { A }\).
2. By considering matrix multiplication, determine whether \(T _ { A }\) followed by \(T _ { B }\) is the same transformation as \(T _ { B }\) followed by \(T _ { A }\). Transformations R and S are each defined as being the result of successive transformations, as specified in the table.

   Transformation	First transformation	followed by
   R	\(\mathrm { T } _ { \mathrm { A } }\) followed by \(\mathrm { T } _ { \mathrm { B } }\)	\(\mathrm { T } _ { \mathrm { C } }\)
   S	\(\mathrm { T } _ { \mathrm { A } }\)	\(\mathrm { T } _ { \mathrm { B } }\) followed by \(\mathrm { T } _ { \mathrm { C } }\)

3. Explain, using a property of matrix multiplication, why R and S are the same transformations. A quadrilateral, \(Q\), has vertices \(D , E , F\) and \(G\) in anticlockwise order from \(D\). Under transformation \(\mathrm { R } , Q ^ { \prime }\) s image, \(Q ^ { \prime }\), has vertices \(D ^ { \prime } , E ^ { \prime } , F ^ { \prime }\) and \(G ^ { \prime }\) (where \(D ^ { \prime }\) is the image of \(D\), etc). The area of \(Q\), in suitable units, is 5 . You are given that det \(\mathbf { C } = a ^ { 2 } + 1\) where \(a\) is a real constant.
4. 1. Determine the order of the vertices of \(Q ^ { \prime }\), starting anticlockwise from \(D ^ { \prime }\).
   2. Find, in terms of \(a\), the area of \(Q ^ { \prime }\).
   3. Explain whether the inverse transformation for R exists. Justify your answer.

6 The matrices \(\mathbf { M }\) and \(\mathbf { N }\) are \(\left( \begin{array} { l l } 0 & 1 \\ 1 & 0 \end{array} \right)\) and \(\left( \begin{array} { l l } 2 & 0 \\ 0 & 1 \end{array} \right)\) respectively.

1. In this question you must show detailed reasoning. Determine whether \(\mathbf { M }\) and \(\mathbf { N }\) commute under matrix multiplication.
2. Specify the transformation of the plane associated with each of the following matrices.
   1. M
   2. N
3. State the significance of the result in part (a) for the transformations associated with \(\mathbf { M }\) and \(\mathbf { N }\). [1]
4. Use an algebraic method to show that all lines parallel to the \(x\)-axis are invariant lines of the transformation associated with N.

5 Matrices \(\mathbf { A }\) and \(\mathbf { B }\) are given by \(\mathbf { A } = \left( \begin{array} { r l } - 1 & 0 \\ 0 & 1 \end{array} \right)\) and \(\mathbf { B } = \left( \begin{array} { c c } \frac { 5 } { 13 } & - \frac { 12 } { 13 } \\ \frac { 12 } { 13 } & \frac { 5 } { 13 } \end{array} \right)\).

1. Use \(\mathbf { A }\) and \(\mathbf { B }\) to disprove the proposition: "Matrix multiplication is commutative". Matrix \(\mathbf { B }\) represents the transformation \(\mathrm { T } _ { \mathrm { B } }\).
2. Describe the transformation \(\mathrm { T } _ { \mathrm { B } }\).
3. By considering the inverse transformation of \(\mathrm { T } _ { \mathrm { B } }\), determine \(\mathbf { B } ^ { - 1 }\). Matrix \(\mathbf { C }\) is given by \(\mathbf { C } = \left( \begin{array} { r r } 1 & 0 \\ 0 & - 3 \end{array} \right)\) and represents the transformation \(\mathrm { T } _ { \mathrm { C } }\).
   The transformation \(\mathrm { T } _ { \mathrm { BC } }\) is transformation \(\mathrm { T } _ { \mathrm { C } }\) followed by transformation \(\mathrm { T } _ { \mathrm { B } }\).
   An object shape of area 5 is transformed by \(\mathrm { T } _ { \mathrm { BC } }\) to an image shape \(N\).
4. Determine the area of \(N\).