Composite transformations, area and invariant lines Problem set A

Find the single matrix for each pair of transformations.

1. (a) \(\begin{pmatrix}3 & 0 \\ 1 & 2\end{pmatrix}\) followed by \(\begin{pmatrix}1 & 2 \\ 0 & 1\end{pmatrix}\)
2. (b) Rotation 90° anticlockwise about the origin, followed by reflection in the \(x\)-axis
3. (c) Enlargement with scale factor 2, followed by rotation 90° anticlockwise

Find the area of the image.

1. (a) The unit square under \(\begin{pmatrix}4 & -1 \\ 3 & 8\end{pmatrix}\)
2. (b) The triangle with vertices \((0,0),\ (3,0),\ (0,2)\) under \(\begin{pmatrix}2 & 4 \\ 3 & 0\end{pmatrix}\)
3. (c) A rectangle of area 5 under \(\begin{pmatrix}1.6 & 0 \\ 0 & 1\end{pmatrix}\)

Find the image of each point.

1. (a) \((2, 3)\): enlargement, scale factor 3, then translation by \(\begin{pmatrix}-5 \\ 0\end{pmatrix}\)
2. (b) \((2, 3)\): translation by \(\begin{pmatrix}-5 \\ 0\end{pmatrix}\), then rotation 90° anticlockwise about the origin

Find the equations of the invariant lines through the origin for each matrix.

1. (a) \(\begin{pmatrix}3 & 2 \\ 1 & 4\end{pmatrix}\)
2. (b) \(\begin{pmatrix}1 & 2 \\ 4 & 3\end{pmatrix}\)

The transformation \(T\) has matrix \(\begin{pmatrix}1 & 2 \\ 0 & 3\end{pmatrix}\).

1. (a) Find the eigenvalues of the matrix.
2. (b) Find the equation of the line of invariant points.
3. (c) Find the equation of the other invariant line through the origin, and say how far points on it move from the origin.

Describe the single transformation equal to a rotation of 90° anticlockwise about the origin followed by a reflection in the line \(y = x\).

\(M = \begin{pmatrix}2 & 1 \\ 1 & 2\end{pmatrix}\). Find the area of the image of the unit square under \(M^4\).

Explain why a rotation about the origin through 90° has no invariant lines through the origin, and a rotation through 180° has infinitely many.

Transformation \(T\) is a reflection in the \(x\)-axis, then an enlargement with scale factor 3, then a rotation of 90° clockwise about the origin.

1. (a) Find the single matrix for \(T\).
2. (b) A shape has area 4. Find the area of its image under \(T\).

A logo is made from a triangle with vertices \((0,0),\ (4,0)\) and \((0,4)\), and its images under \(T,\ T^2,\ T^3,\ \ldots\), where \(T\) is an enlargement with scale factor \(\frac{1}{2}\) followed by a rotation of 90° anticlockwise about the origin. (a) Find the matrix of \(T\). (b) Find the area of the first triangle and of its image under \(T\). (c) Find the total area of all the triangles.