Matrix algebra and linear transformationsEdexcel A-Level Further Maths: Revision notes

Matrices of the same size are added or subtracted entry by entry, and multiplying by a scalar multiplies every entry: $2\mathbf{B}$ doubles each entry. Two matrices are conformable for the product $\mathbf{AB}$ if the number of columns of $\mathbf{A}$ equals the number of rows of $\mathbf{B}$. An $m\times n$ matrix times an $n\times p$ matrix gives an $m\times p$ matrix, and each entry is a row of $\mathbf{A}$ times a column of $\mathbf{B}$. For $\mathbf{A}=\begin{pmatrix}2&1\\0&3\end{pmatrix}$ and $\mathbf{B}=\begin{pmatrix}1&-1\\2&0\end{pmatrix}$: $\mathbf{AB}=\begin{pmatrix}4&-2\\6&0\end{pmatrix}$ but $\mathbf{BA}=\begin{pmatrix}2&-2\\4&2\end{pmatrix}$. Matrix multiplication is associative ($\mathbf{A}(\mathbf{BC})=(\mathbf{AB})\mathbf{C}$) but not commutative.

The zero matrix $\mathbf{0}$ has every entry zero: $\mathbf{A}+\mathbf{0}=\mathbf{A}$ and $\mathbf{A}\mathbf{0}=\mathbf{0}$. The identity matrix $\mathbf{I}$ has $1$s on the leading diagonal and $0$s elsewhere. For $2\times2$, $\mathbf{I}=\begin{pmatrix}1&0\\0&1\end{pmatrix}$, and $\mathbf{AI}=\mathbf{IA}=\mathbf{A}$ for any conformable $\mathbf{A}$. The identity represents the transformation that does nothing. Powers follow as normal: $\mathbf{A}^2=\mathbf{AA}$, and $\mathbf{A}^0=\mathbf{I}$. For the rotation $\mathbf{M}$ through $60^\circ$, $\mathbf{M}^6=\mathbf{I}$ because six rotations of $60^\circ$ make a full turn.

A matrix $\begin{pmatrix}a&b\\ c&d\end{pmatrix}$ maps the point $(x,y)$ to $(ax+by,\,cx+dy)$. Its columns are the images of the unit vectors: $(1,0)\mapsto(a,c)$ and $(0,1)\mapsto(b,d)$. So to find a matrix, work out where $(1,0)$ and $(0,1)$ go and write them as columns.

• Reflection in the $x$-axis: $\begin{pmatrix}1&0\\0&-1\end{pmatrix}$; in the $y$-axis: $\begin{pmatrix}-1&0\\0&1\end{pmatrix}$.
• Reflection in $y=x$: $\begin{pmatrix}0&1\\1&0\end{pmatrix}$; in $y=-x$: $\begin{pmatrix}0&-1\\-1&0\end{pmatrix}$.
• Rotation through $\theta$ anticlockwise about the origin: $\begin{pmatrix}\cos\theta&-\sin\theta\\ \sin\theta&\cos\theta\end{pmatrix}$; use $-\theta$ for clockwise.
• Stretch scale factor $k$ parallel to the $x$-axis: $\begin{pmatrix}k&0\\0&1\end{pmatrix}$; parallel to the $y$-axis: $\begin{pmatrix}1&0\\0&k\end{pmatrix}$.
• Enlargement, scale factor $k$, centre the origin: $\begin{pmatrix}k&0\\0&k\end{pmatrix}$.

If $\mathbf{B}$ is applied first and then $\mathbf{A}$, the point $\mathbf{x}$ becomes $\mathbf{A}(\mathbf{Bx})=(\mathbf{AB})\mathbf{x}$. So $\mathbf{AB}$ represents $\mathbf{B}$ followed by $\mathbf{A}$: the first transformation is on the right, next to the point. Example: reflection in $y=x$, $\begin{pmatrix}0&1\\1&0\end{pmatrix}$, followed by a rotation through $90^\circ$ anticlockwise, $\begin{pmatrix}0&-1\\1&0\end{pmatrix}$, is $\begin{pmatrix}0&-1\\1&0\end{pmatrix}\begin{pmatrix}0&1\\1&0\end{pmatrix}=\begin{pmatrix}-1&0\\0&1\end{pmatrix}$: a reflection in the $y$-axis. Checking with $(3,2)\to(2,3)\to(-3,2)$ confirms it. Reversing the order gives $\begin{pmatrix}1&0\\0&-1\end{pmatrix}$, a reflection in the $x$-axis, which shows that order matters.

A $3\times3$ matrix transforms points $(x,y,z)$, and its columns are the images of $(1,0,0)$, $(0,1,0)$ and $(0,0,1)$. The 3-D transformations in this specification are reflections in a coordinate plane and rotations about a coordinate axis.

• Reflection in $x=0$: $\begin{pmatrix}-1&0&0\\0&1&0\\0&0&1\end{pmatrix}$; in $y=0$ and $z=0$, the $-1$ moves to the second or third diagonal entry.
• Rotation about the $x$-axis through $\theta$ (anticlockwise looking towards the origin): $\begin{pmatrix}1&0&0\\0&\cos\theta&-\sin\theta\\0&\sin\theta&\cos\theta\end{pmatrix}$.
• Rotation about the $z$-axis: $\begin{pmatrix}\cos\theta&-\sin\theta&0\\ \sin\theta&\cos\theta&0\\0&0&1\end{pmatrix}$; the $y$-axis one is $\begin{pmatrix}\cos\theta&0&\sin\theta\\0&1&0\\-\sin\theta&0&\cos\theta\end{pmatrix}$. The axis you rotate about is fixed, so its row and column are those of the identity. Successive 3-D transformations multiply in the same right-to-left order as in 2-D.