IGCSE Math B 5.G Combined transformations Topic Practice

Triangles A,D and F are shown on the grid.

In this question, ignore any labelling on the triangles.

Triangle E is transformed to triangle B under the transformation represented by matrix $\mathbf N$.
Find the matrix $\mathbf N$.

\text{If }A=\begin{pmatrix}a&b\c&d\end{pmatrix},\text{ then }A^{-1}=\frac{1}{ad-bc}\begin{pmatrix}d&-b\-c&a\end{pmatrix}.

The points with coordinates (3,1),(4,3),(6,3) and (6,1) are the vertices of a quadrilateral Q.

Find the matrix that represents the transformation that maps quadrilateral S onto quadrilateral Q.

The diagram shows kite A.
Kite B is the image of kite A under a reflection in the line with equation y=-1.

$$M=\begin{pmatrix}0&-1\\-2&0\end{pmatrix}, \qquad N=\begin{pmatrix}0&-1\\1&0\end{pmatrix}.$$
Kite A is transformed to kite D under the combined transformation with matrix MN.
On the grid opposite, draw and label kite D.

The points with coordinates (1,1),(3,1) and (4,3) are the vertices of triangle A.

Triangle A is transformed to triangle C under the transformation with matrix N.

Find matrix N.

Triangles A and B are drawn on the grid opposite.

Triangle D is the image of triangle A under the transformation with matrix N.
Find the matrix N.

Only use this grid if you need to redraw your triangles.

Triangle A and triangle B are drawn on the grid opposite.

Triangle B is transformed to triangle E under the transformation with matrix $\mathbf P$, where
$$\mathbf P=\begin{pmatrix}-k&1\\k-3&0\end{pmatrix}.$$
Triangle E is transformed to triangle F under the transformation with matrix $\mathbf Q$, where
$$\mathbf Q=\begin{pmatrix}k&1\\k^2-1&k\end{pmatrix}.$$
Triangle F is the image of triangle B under the matrix $\mathbf N$.
Given that the determinant of $\mathbf N$ is 2, find the coordinates of the vertices of triangle F.

$$\det\begin{pmatrix}a&b\\c&d\end{pmatrix}=ad-bc.$$

$$A=\begin{pmatrix}3&4\\-1&2\end{pmatrix}, \qquad B=\begin{pmatrix}3&-1\\2&-2\end{pmatrix}.$$
The transformation with matrix A is equivalent to the transformation with matrix B followed by the transformation with matrix C, where C is a $2\times2$ matrix.
Find matrix C.

The inverse of matrix $\begin{pmatrix}a&b\\c&d\end{pmatrix}$ is
$$\frac1{ad-bc}\begin{pmatrix}d&-b\\-c&a\end{pmatrix}.$$

$$\mathbf A=\begin{pmatrix}2k^2&k-9\\-3k&k+1\end{pmatrix}, \qquad \mathbf B=\begin{pmatrix}1&-5\\3&k\end{pmatrix}.$$
The determinant of matrix $\mathbf A$ is equal to the determinant of matrix $\mathbf B$.

The transformation with matrix $\mathbf C$, where $\mathbf C$ is a $2\times2$ matrix, is equivalent to the transformation with matrix $\mathbf A$ followed by the transformation with matrix $\mathbf B$.
Given that k is positive, find matrix $\mathbf C$.

$$\text{Determinant of matrix }\begin{pmatrix}a&b\\c&d\end{pmatrix}=ad-bc.$$

Triangle A and triangle D are drawn on the grid opposite.
Triangle A is transformed to triangle B under a rotation of $180^\circ$ about (3,1).
Triangle A is transformed to triangle C under a reflection in the line y=-0.5.
Triangle D is transformed to triangle E under the transformation with matrix
$$\mathbf M=\begin{pmatrix}-2&0\\0&-2\end{pmatrix}.$$

Triangle A is transformed to triangle F under a rotation of $90^\circ$ anticlockwise about the origin.
Triangle F is transformed to triangle G under the matrix $\begin{pmatrix}7-2k&2\\2&4\end{pmatrix}$.
Triangle G is the image of triangle A under the matrix $\mathbf N$.
Given that $\mathbf N^{-1}=\frac1{10}\begin{pmatrix}-1&3\\-2&1\end{pmatrix}$, find the value of k.

$$\text{Inverse of }\begin{pmatrix}a&b\\c&d\end{pmatrix}\text{ is }\frac1{ad-bc}\begin{pmatrix}d&-b\\-c&a\end{pmatrix}.$$