Three-dimensional transformations and 3x3 matricesEdexcel International A Level Further Maths: Revision notes

A $3\times3$ matrix represents a linear transformation of column vectors in three dimensions. Its columns are the images of $\mathbf{i}=(1,0,0)$, $\mathbf{j}=(0,1,0)$ and $\mathbf{k}=(0,0,1)$, so you can write down the matrix from where the unit vectors go. Standard matrices:

• reflection in $z=0$: $\operatorname{diag}(1,1,-1)$; in $x=0$: $\operatorname{diag}(-1,1,1)$;
• anticlockwise rotation $\theta$ about the $z$-axis: $\begin{pmatrix} \cos\theta & -\sin\theta & 0 \\ \sin\theta & \cos\theta & 0 \\ 0 & 0 & 1 \end{pmatrix}$, and about the $x$-axis: $\begin{pmatrix} 1 & 0 & 0 \\ 0 & \cos\theta & -\sin\theta \\ 0 & \sin\theta & \cos\theta \end{pmatrix}$;
• stretch of scale factor $a$, $b$, $c$ in the three axis directions: $\operatorname{diag}(a,b,c)$. In 2D the same ideas apply with $2\times2$ matrices, and the 3D versions extend them.

If $\mathbf{A}$ is applied first and $\mathbf{B}$ second, the combined transformation has matrix $\mathbf{BA}$: $\mathbf{AB}$ means $\mathbf{B}$ followed by $\mathbf{A}$. Matrix multiplication is not commutative, so the order matters. Example: $\mathbf{P}$ (rotation $90^\circ$ about $z$) followed by $\mathbf{Q}$ (reflection in $x=0$) is $\mathbf{QP}=\begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{pmatrix}$, a reflection in the plane $x=y$. $\mathbf{PQ}$ is a different matrix, $\begin{pmatrix} 0 & -1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & 1 \end{pmatrix}$.

The transpose $\mathbf{A}^{\mathrm{T}}$ interchanges rows and columns, so $(\mathbf{A}^{\mathrm{T}})_{ij}=\mathbf{A}_{ji}$. Key results: $(\mathbf{AB})^{\mathrm{T}}=\mathbf{B}^{\mathrm{T}}\mathbf{A}^{\mathrm{T}},\qquad(\mathbf{A}^{\mathrm{T}})^{\mathrm{T}}=\mathbf{A},\qquad\det\mathbf{A}^{\mathrm{T}}=\det\mathbf{A}.$ Example: $\mathbf{A}=\begin{pmatrix} 1 & 2 & 0 \\ 3 & 4 & -1 \\ 2 & 0 & 1 \end{pmatrix}$ has $\mathbf{A}^{\mathrm{T}}=\begin{pmatrix} 1 & 3 & 2 \\ 2 & 4 & 0 \\ 0 & -1 & 1 \end{pmatrix}$. For a rotation matrix, $\mathbf{R}^{\mathrm{T}}=\mathbf{R}^{-1}$.

Expand a $3\times3$ determinant along a row or column using the sign pattern $\begin{pmatrix} + & - & + \\ - & + & - \\ + & - & + \end{pmatrix}$: $\begin{vmatrix} a & b & c \\ d & e & f \\ g & h & i \end{vmatrix}=a(ei-fh)-b(di-fg)+c(dh-eg).$ A matrix is singular if $\det\mathbf{A}=0$ (no inverse) and non-singular otherwise. Example: $\begin{pmatrix} 1 & 2 & 0 \\ 3 & k & -1 \\ 2 & 0 & 1 \end{pmatrix}$ has determinant $k-10$, so it is singular when $k=10$. Also $\det(\mathbf{AB})=\det\mathbf{A}\det\mathbf{B}$.

For a non-singular $3\times3$ matrix, $\mathbf{A}^{-1}=\frac{1}{\det\mathbf{A}}\operatorname{adj}\mathbf{A}$. Method:

1. Find $\det\mathbf{A}$.
2. Find the matrix of minors, apply the $+-+$ signs to get cofactors.
3. Transpose to obtain the adjugate.
4. Divide by $\det\mathbf{A}$ and check $\mathbf{AA}^{-1}=\mathbf{I}$. For products, $(\mathbf{AB})^{-1}=\mathbf{B}^{-1}\mathbf{A}^{-1}$. The inverse of a transformation undoes it: if $T$ is $\mathbf{A}$ then $\mathbf{B}$, then $T^{-1}$ undoes $\mathbf{B}$ first and then $\mathbf{A}$, so its matrix is $\mathbf{A}^{-1}\mathbf{B}^{-1}$. A reflection is its own inverse; the inverse of an anticlockwise rotation is the clockwise rotation about the same axis.