Composite transformations, area and invariant lines Problem set B Answers

1. (a) \(\begin{pmatrix}2 & -1 \\ 8 & -1\end{pmatrix}\)
2. (b) \(\begin{pmatrix}0 & 1 \\ 1 & 0\end{pmatrix}\)
3. (c) \(\begin{pmatrix}0 & 3 \\ -3 & 0\end{pmatrix}\)

1. (a) 26
2. (b) 48
3. (c) 20

1. (a) \((6, -5)\)
2. (b) \((2, -3)\)

1. (a) \(y = x\) and \(y = -2x\)
2. (b) \(y = \frac{2}{3}x\) and \(y = x\)

1. (a) \(\lambda = 1 \text{ or } 4\)
2. (b) \(y = -x\)
3. (c) \(x = 0\); points move 4 times as far from the origin.

\(\begin{pmatrix}0 & -1 \\ 1 & 0\end{pmatrix}\begin{pmatrix}-1 & 0 \\ 0 & 1\end{pmatrix} = \begin{pmatrix}0 & -1 \\ -1 & 0\end{pmatrix}\): a reflection in the line \(y = -x\).

\(\det M = 5\), so the area is \(5^3 = 125\).

A 120° turn changes the direction of every vector, so no line maps to itself (no real eigenvalues). An enlargement sends every vector to 3 times itself, which is on the same line: every line through the origin is invariant (eigenvalue 3).

1. (a) \(\begin{pmatrix}0 & 3 \\ 1 & 0\end{pmatrix}\)
2. (b) 18

(a) \(\begin{pmatrix}0 & \frac{1}{3} \\ -\frac{1}{3} & 0\end{pmatrix}\). (b) 12 and \(\frac{4}{3}\) (\(\det T = \frac{1}{9}\)). (c) A geometric series: \(\dfrac{12}{1 - \frac{1}{9}} = \dfrac{27}{2}\).