CAIE A-Level Further Mathematics 1.4.4 Matrix Transformations

Practise identifying planar transformations from 2 × 2 matrices, composing them in the correct order and using determinants to track changes in area.

[Maximum number: 5]

Let k be a constant. The matrices A, B and C are given by

\mathbf{A}=\left(\begin{array}{lll} 1 & k & 3 \\ 2 & 1 & 3 \\ 3 & 2 & 5 \end{array}\right), \quad \mathbf{B}=\left(\begin{array}{rr} 0 & -2 \\ -1 & 3 \\ 0 & 0 \end{array}\right) \quad \text { and } \quad \mathbf{C}=\left(\begin{array}{rrr} -2 & -1 & 1 \\ 1 & 1 & 3 \end{array}\right)

It is given that A is singular.

The matrices D, E and F represent geometrical transformations in the x-y plane.
- D represents an enlargement, centre the origin.
- E represents a stretch parallel to the x-axis.
- F represents a reflection in the line y=x.

Given that C A B=D-9 E F, find D, E and F.

$\mathbf{D}=\left(\begin{array}{ll}\alpha 0 \alpha\end{array}\right)$

B1

$\mathbf{E}=\left(\begin{array}{ll}\beta 0 1\end{array}\right)$

B1

$\mathbf{F}=\left(\begin{array}{ll}0 1 0\end{array}\right)$

B1

$\left(\begin{array}{cc}3 -7 -9 3\end{array}\right)=\left(\begin{array}{ll}\alpha 0 \alpha\end{array}\right)-9\left(\begin{array}{ll}0 \beta 1 0\end{array}\right)$

M1

Setting up simultaneous equations using their D and
E.

$\mathbf{D}=\left(\begin{array}{ll}3 0 3\end{array}\right) \quad \mathbf{E}=\left(\begin{array}{ll}\frac{7}{9} 0 1\end{array}\right)$

A1

Condone $\alpha=3, \beta=\frac{7}{9}$ if it is clear that they refer to
the correct matrices.

5

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1.4.4—Matrix transformations question 1