Determinants Problem set A

Find the determinant of each matrix.

1. (a) \(\begin{pmatrix}5 & 2 \\ 3 & 4\end{pmatrix}\)
2. (b) \(\begin{pmatrix}-2 & 7 \\ 1 & 3\end{pmatrix}\)
3. (c) \(\begin{pmatrix}6 & -4 \\ -3 & 2\end{pmatrix}\)
4. (d) \(\begin{pmatrix}0 & 5 \\ -2 & 9\end{pmatrix}\)
5. (e) \(\begin{pmatrix}0.5 & 3 \\ 2 & 8\end{pmatrix}\)
6. (f) \(\begin{pmatrix}-4 & -1 \\ 6 & -3\end{pmatrix}\)

Find the area of each triangle.

1. (a) Vertices \((0,0),\ (6,1),\ (2,5)\)
2. (b) Vertices \((2,1),\ (7,4),\ (3,8)\)
3. (c) Vertices \((-1,2),\ (4,0),\ (1,6)\)

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

1. (a) \(\begin{pmatrix}k & 3 \\ 4 & k+1\end{pmatrix}\)
2. (b) \(\begin{pmatrix}k-1 & 4 \\ 2 & k+1\end{pmatrix}\)
3. (c) \(\begin{pmatrix}3 & k \\ k & 12\end{pmatrix}\)

Use your calculator to find the determinant of each matrix.

1. (a) \(\begin{pmatrix}2 & 0 & 1 \\ -1 & 3 & 4 \\ 5 & 2 & -2\end{pmatrix}\)
2. (b) \(\begin{pmatrix}1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9\end{pmatrix}\)

\(A\) is the point \((3, 2)\). 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 \((1, 2)\), \((3, 6)\) and \((6, 12)\) lie on one straight line.

A quadrilateral has vertices \((0,0),\ (5,1),\ (6,5)\) and \((1,4)\), in order. Show that it is a parallelogram and find its area.

\(A = \begin{pmatrix}2 & 1 \\ 3 & 4\end{pmatrix}\) and \(B = \begin{pmatrix}1 & -2 \\ 5 & 0\end{pmatrix}\). Find \(\det A\), \(\det B\) and \(\det(AB)\). What do you notice?

For \(M = \begin{pmatrix}a & b \\ c & d\end{pmatrix}\), show that \(\det(kM) = k^2 \det M\). Explain this using area.