Determinants Problem set B

Find the determinant of each matrix.

1. (a) \(\begin{pmatrix}4 & 3 \\ 2 & 7\end{pmatrix}\)
2. (b) \(\begin{pmatrix}-3 & 5 \\ 2 & 4\end{pmatrix}\)
3. (c) \(\begin{pmatrix}8 & -6 \\ -4 & 3\end{pmatrix}\)
4. (d) \(\begin{pmatrix}0 & -4 \\ 3 & 6\end{pmatrix}\)
5. (e) \(\begin{pmatrix}2.5 & 2 \\ 3 & 4\end{pmatrix}\)
6. (f) \(\begin{pmatrix}-5 & -2 \\ 4 & -3\end{pmatrix}\)

Find the area of each triangle.

1. (a) Vertices \((0,0),\ (7,2),\ (3,6)\)
2. (b) Vertices \((1,2),\ (6,3),\ (2,7)\)
3. (c) Vertices \((-2,1),\ (3,-1),\ (1,5)\)

Find the values of \(k\) for which each matrix has determinant zero.

1. (a) \(\begin{pmatrix}k & 5 \\ 2 & k + 3\end{pmatrix}\)
2. (b) \(\begin{pmatrix}k + 3 & 2 \\ 8 & k - 3\end{pmatrix}\)
3. (c) \(\begin{pmatrix}3 & k \\ k & 27\end{pmatrix}\)

Use your calculator to find the determinant of each matrix.

1. (a) \(\begin{pmatrix}3 & 1 & 0 \\ 2 & -1 & 4 \\ 1 & 5 & -2\end{pmatrix}\)
2. (b) \(\begin{pmatrix}1 & 3 & 5 \\ 2 & 4 & 6 \\ 3 & 5 & 7\end{pmatrix}\)

\(A\) is the point \((5, 2)\). Give the coordinates of a point \(B\) so that the determinant of the matrix with columns \(\overrightarrow{OA}\) and \(\overrightarrow{OB}\) is:

1. (a) zero
2. (b) negative
3. (c) 12

Use a determinant to show that the points \((1, 4)\), \((4, 6)\) and \((7, 8)\) lie on one straight line.

A quadrilateral has vertices \((0,0),\ (6,2),\ (8,6)\) and \((2,4)\), in order. Show that it is a parallelogram and find its area.

\(A = \begin{pmatrix}3 & 1 \\ 2 & 5\end{pmatrix}\) and \(B = \begin{pmatrix}2 & -3 \\ 4 & 1\end{pmatrix}\). Find \(\det A\), \(\det B\) and \(\det(AB)\). What do you notice?

For \(N = \begin{pmatrix}p & q \\ r & s\end{pmatrix}\), show that \(\det(3N) = 9 \det N\). Explain this using area.