Find invariant points

1

1. Give full details of the geometrical transformation in the \(x - y\) plane represented by the matrix \(\left( \begin{array} { l l } 6 & 0 \\ 0 & 6 \end{array} \right)\). Let \(\mathbf { A } = \left( \begin{array} { l l } 3 & 4 \\ 2 & 2 \end{array} \right)\).
2. The triangle \(D E F\) in the \(x - y\) plane is transformed by \(\mathbf { A }\) onto triangle \(P Q R\). Given that the area of triangle \(D E F\) is \(13 \mathrm {~cm} ^ { 2 }\), find the area of triangle \(P Q R\).
3. Find the matrix \(\mathbf { B }\) such that \(\mathbf { A B } = \left( \begin{array} { l l } 6 & 0 \\ 0 & 6 \end{array} \right)\).
4. Show that the origin is the only invariant point of the transformation in the \(x - y\) plane represented by \(\mathbf { A }\).

1. A point \(P\) is reflected in the line \(y = k x\), where \(k\) is a constant. It is then rotated anticlockwise about \(O\) through an acute angle \(\theta\), where \(\cos \theta = 0 \cdot 8\). The resulting transformation matrix is given by \(T\), where

$$T = \frac { 1 } { 85 } \left[ \begin{array} { r r } - 84 & - 13 \\ - 13 & 84 \end{array} \right]$$

1. Determine the value of \(k\).
   Find the invariant points of \(T\).

5. $$\mathbf { A } = \left( \begin{array} { r r } - \frac { 1 } { 2 } & - \frac { \sqrt { 3 } } { 2 } \\ \frac { \sqrt { 3 } } { 2 } & - \frac { 1 } { 2 } \end{array} \right)$$

1. Describe fully the single geometrical transformation \(U\) represented by the matrix \(\mathbf { A }\). The transformation \(V\), represented by the \(2 \times 2\) matrix \(\mathbf { B }\), is a reflection in the line \(y = - x\)
2. Write down the matrix \(\mathbf { B }\). Given that \(U\) followed by \(V\) is the transformation \(T\), which is represented by the matrix \(\mathbf { C }\), (c) find the matrix \(\mathbf { C }\).
3. Show that there is a real number \(k\) for which the point \(( 1 , k )\) is invariant under \(T\).

   V349 SIHI NI IMIMM ION OC

   VJYV SIHIL NI LIIIM ION OO

   VJYV SIHIL NI JIIYM ION OC

3 The matrix \(\mathbf { A } = \left[ \begin{array} { l l } 1 & 2 \\ 0 & 1 \end{array} \right]\) represents a transformation.
Which one of the points below is an invariant point under this transformation?
Circle your answer. \(( 1,1 )\) \(( 0,2 )\) \(( 3,0 )\) \(( 2,1 )\)