VT · Matrix Transformations

Non-calculator

In each set, one thing changes and everything else stays the same. Work them out in order and look for the pattern — the last line tells you what to notice.

①Apply $\begin{pmatrix}0&-1\\1&0\end{pmatrix}$ (rotate $90^{\circ}$ acw)changing: the point
Find the image of each point.
(2, 0)
=
(0, 3)
=
(2, 3)
=
(-1, 4)
=
The rule sends $(x,y)\to(-y,x)$. Watch how a quarter-turn moves each point.
②Same point $(3, 2)$, different matrixchanging: the transformation
Find the image of $(3, 2)$ under each matrix.
\begin{pmatrix}1&0\\0&-1\end{pmatrix}
=
\begin{pmatrix}-1&0\\0&1\end{pmatrix}
=
\begin{pmatrix}0&1\\1&0\end{pmatrix}
=
\begin{pmatrix}2&0\\0&2\end{pmatrix}
=
The point stays the same; only the transformation changes. Match each matrix to what it does.
③Enlargements, centre Ochanging: the scale factor
The matrix enlarges a shape of area $4\text{ cm}^2$. Find the image area.
k=2
=
k=3
=
k=4
=
k=5
=
The area scale factor is $k^2$, not $k$. Track how fast the area grows.

Answers · Matrix Transformations

Variation practice
① Apply $\begin{pmatrix}0&-1\\1&0\end{pmatrix}$ (rotate $90^{\circ}$ acw)
(2, 0): $(0,\ 2)$(0, 3): $(-3,\ 0)$(2, 3): $(-3,\ 2)$(-1, 4): $(-4,\ -1)$
② Same point $(3, 2)$, different matrix
\begin{pmatrix}1&0\\0&-1\end{pmatrix}: $(3,\ -2)$\begin{pmatrix}-1&0\\0&1\end{pmatrix}: $(-3,\ 2)$\begin{pmatrix}0&1\\1&0\end{pmatrix}: $(2,\ 3)$\begin{pmatrix}2&0\\0&2\end{pmatrix}: $(6,\ 4)$
③ Enlargements, centre O
k=2: $16\text{ cm}^2$k=3: $36\text{ cm}^2$k=4: $64\text{ cm}^2$k=5: $100\text{ cm}^2$
mathedup.co.uk · sheet E4PQ