Matrices and linear transformations – further maths test to download as PDF

\(\mathbf A=\begin{pmatrix}2&-1\\3&4\end{pmatrix}\) and \(\mathbf B=\begin{pmatrix}1&0\\-2&5\end{pmatrix}\).

1. Find \(\mathbf{AB}\). [2]
2. Find \(\det\mathbf A\), \(\det\mathbf B\) and \(\det(\mathbf{AB})\), and verify that \(\det(\mathbf{AB})=\det\mathbf A\det\mathbf B\). [2]

\(\mathbf N=\begin{pmatrix}k&1&0\\2&k&1\\0&1&1\end{pmatrix}\), where \(k\) is a constant.

1. Show that \(\det\mathbf N=k^2-k-2\). [2]
2. Find the values of \(k\) for which \(\mathbf N\) is singular. [1]
3. When \(k=3\), find the volume of the image of a cube of side 2 under \(\mathbf N\). [1]

\(T_1\) is a rotation of \(90^\circ\) clockwise about \(O\) and \(T_2\) is a reflection in the \(y\)-axis.

1. Write down the matrices of \(T_1\) and \(T_2\). [2]
2. Find the matrix of \(T_1\) followed by \(T_2\) and describe this single transformation. [2]

The matrix \(\mathbf M=\begin{pmatrix}2&1\\2&3\end{pmatrix}\) represents a transformation \(T\).

1. Show that every point on the line \(y=-x\) is invariant under \(T\). [1]
2. Find the other invariant line of \(T\) that passes through the origin. [2]
3. Explain whether this second line is a line of invariant points. [1]

On a map, a triangular field has vertices \(O(0,0)\), \(P(4,0)\) and \(Q(0,2)\), in units of 100 m. A software error transforms the map with the matrix \(\mathbf M=\begin{pmatrix}2&1\\1&3\end{pmatrix}\).

1. Find the images of \(P\) and \(Q\). [1]
2. Find the area of the field and the area of its image on the transformed map. [2]
3. A tree appears at \((5,5)\) on the transformed map. Find its true position. [1]