Determinants Problem set B Answers

1. (a) 22
2. (b) -22
3. (c) 0
4. (d) 12
5. (e) 4
6. (f) 23

1. (a) 18
2. (b) 12
3. (c) 13

1. (a) \(k = 2\) or \(k = -5\)
2. (b) \(k = \pm 5\)
3. (c) \(k = \pm 9\)

1. (a) −46
2. (b) 0

1. (a) Any point on the line through O and A, for example \((10, 4)\).
2. (b) Any point clockwise from A, for example \((2, 0)\), which gives −4.
3. (c) Any point with \(5y - 2x = 12\), for example \((4, 4)\) or \((-6, 0)\).

From \((1,4)\) the two vectors are \(\begin{pmatrix}3 \\ 2\end{pmatrix}\) and \(\begin{pmatrix}6 \\ 4\end{pmatrix}\). \(3(4) - 6(2) = 0\): no area, so the points are in line.

\(\begin{pmatrix}6 \\ 2\end{pmatrix} + \begin{pmatrix}2 \\ 4\end{pmatrix} = \begin{pmatrix}8 \\ 6\end{pmatrix}\), so it is a parallelogram. Area \(= 6(4) - 2(2) = 20\).

\(\det A = 13,\; \det B = 14,\; AB = \begin{pmatrix}10 & -8 \\ 24 & -1\end{pmatrix},\; \det(AB) = 182\). \(\det(AB) = \det A \times \det B\).

\(\det(3N) = (3p)(3s) - (3q)(3r) = 9(ps - qr)\). Both vectors are 3 times as long, so the area is 9 times as big.