Geometric interpretation of matrices

9. $$\mathbf { M } = \left( \begin{array} { r r } 3 & 4 \\ 2 & - 5 \end{array} \right)$$

1. Find \(\operatorname { det } \mathbf { M }\). The transformation represented by \(\mathbf { M }\) maps the point \(S ( 2 a - 7 , a - 1 )\), where \(a\) is a constant, onto the point \(S ^ { \prime } ( 25 , - 14 )\).
2. Find the value of \(a\). The point \(R\) has coordinates \(( 6,0 )\). Given that \(O\) is the origin,
3. find the area of triangle \(O R S\). Triangle \(O R S\) is mapped onto triangle \(O R ^ { \prime } S ^ { \prime }\) by the transformation represented by \(\mathbf { M }\).
4. Find the area of triangle \(O R ^ { \prime } S ^ { \prime }\). Given that $$\mathbf { A } = \left( \begin{array} { r r } 0 & - 1 \\ 1 & 0 \end{array} \right)$$
5. describe fully the single geometrical transformation represented by \(\mathbf { A }\). The transformation represented by \(\mathbf { A }\) followed by the transformation represented by \(\mathbf { B }\) is equivalent to the transformation represented by \(\mathbf { M }\).
6. Find B.

5 The matrices \(\mathbf { A }\) and \(\mathbf { B }\) are defined by $$\mathbf { A } = \left[ \begin{array} { c c } k & k \\ k & - k \end{array} \right] , \quad \mathbf { B } = \left[ \begin{array} { c c } - k & k \\ k & k \end{array} \right]$$ where \(k\) is a constant.

1. Find, in terms of \(k\) :
   1. \(\mathbf { A } + \mathbf { B }\);
   2. \(\mathbf { A } ^ { 2 }\).
2. Show that \(( \mathbf { A } + \mathbf { B } ) ^ { 2 } = \mathbf { A } ^ { 2 } + \mathbf { B } ^ { 2 }\).
3. It is now given that \(k = 1\).
   1. Describe the geometrical transformation represented by the matrix \(\mathbf { A } ^ { 2 }\).
   2. The matrix \(\mathbf { A }\) represents a combination of an enlargement and a reflection. Find the scale factor of the enlargement and the equation of the mirror line of the reflection.