Sixth Term Examination Paper (STEP) Pure Mathematics: Vectors and Matrices Syllabus

State the conditions required for matrix addition and matrix multiplication to be defined.

Addition: matrices must have the same dimensions. Multiplication $AB$: the number of columns of $A$ must equal the number of rows of $B$; if $A$ is $m\times n$ and $B$ is $n\times p$, then $AB$ is $m\times p$.

Is matrix multiplication commutative? State the relevant property and the transpose rule for products.

No: in general $AB \neq BA$. Matrix multiplication is associative and distributive. The transpose of a product reverses order: $(AB)^{T} = B^{T}A^{T}$.

What is the identity matrix and its defining property?

The identity matrix $I$ has $1$s on the main diagonal and $0$s elsewhere. For any compatible matrix $A$, $AI = IA = A$.

Give the formula for the determinant of a $2\times 2$ matrix $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$.

$\det = ad - bc$.

State the cofactor expansion for the determinant of a $3\times 3$ matrix along the first row.

For $\begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix}$: $\det = a(ei - fh) - b(di - fg) + c(dh - eg)$.

What does a determinant of zero tell you about a square matrix?

The matrix is singular (non-invertible); its columns/rows are linearly dependent. As a transformation it collapses space onto a lower dimension (zero area/volume scaling).

State the multiplicative property of determinants for a product of square matrices.

$\det(AB) = \det(A)\det(B)$. Also $\det(A^{-1}) = \dfrac{1}{\det(A)}$ and $\det(A^{T})=\det(A)$.

Give the formula for the inverse of a $2\times 2$ matrix $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$.

$A^{-1} = \dfrac{1}{ad-bc}\begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$, valid when $ad-bc \neq 0$.

Describe the general method for finding the inverse of an invertible $3\times 3$ matrix using cofactors.

$A^{-1} = \dfrac{1}{\det(A)}\,\mathrm{adj}(A)$, where the adjugate is the transpose of the cofactor matrix (each cofactor $C_{ij} = (-1)^{i+j}M_{ij}$, with $M_{ij}$ the minor).

How can the matrix inverse be used to solve the simultaneous system $A\vec{x} = \vec{b}$?

If $A$ is invertible, $\vec{x} = A^{-1}\vec{b}$. This gives the unique solution when $\det(A)\neq 0$.

What does $\det(A)=0$ imply about the solutions of the linear system $A\vec{x}=\vec{b}$?

The system has either no solutions (inconsistent) or infinitely many solutions; it does not have a unique solution. The associated geometry is parallel or coincident planes/lines.

Which matrix represents a rotation about the origin through angle $\theta$ anticlockwise in the plane?

$\begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}$.

Which matrix represents a reflection in the line $y = x\tan\theta$ (line through the origin at angle $\theta$)?

$\begin{pmatrix} \cos 2\theta & \sin 2\theta \\ \sin 2\theta & -\cos 2\theta \end{pmatrix}$.

Give the matrices for an enlargement scale factor $k$ about the origin and for reflection in the $x$-axis.

Enlargement: $\begin{pmatrix} k & 0 \\ 0 & k \end{pmatrix}$. Reflection in the $x$-axis: $\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$.

What does the determinant of a $2\times 2$ transformation matrix represent geometrically?

Its absolute value is the area scale factor of the transformation; a negative sign indicates that orientation is reversed (a reflection is involved).

If transformation $A$ is applied first and then transformation $B$, what single matrix represents the composite?

The product $BA$ (the second transformation's matrix on the left). Order matters because matrix multiplication is not commutative.

Define an invariant point of a linear transformation represented by matrix $M$.

A point with position vector $\vec{r}$ such that $M\vec{r} = \vec{r}$; it is mapped to itself. The origin is always invariant for a linear transformation.

What is an invariant line of a transformation, and how does it differ from a line of invariant points?

An invariant line is mapped onto itself as a whole ($M$ sends points of the line to other points of the same line). A line of invariant points has every individual point fixed, which is stronger.

Define an eigenvalue and eigenvector of a square matrix $A$.

A nonzero vector $\vec{v}$ is an eigenvector with eigenvalue $\lambda$ if $A\vec{v} = \lambda\vec{v}$. The direction of $\vec{v}$ is preserved (scaled by $\lambda$) under $A$.

State the characteristic equation used to find the eigenvalues of $A$.

$\det(A - \lambda I) = 0$. Its roots are the eigenvalues.

Once an eigenvalue $\lambda$ is known, how do you find a corresponding eigenvector?

Solve $(A - \lambda I)\vec{v} = \vec{0}$ for a nonzero vector $\vec{v}$; the solution space gives the eigenvectors for that $\lambda$.

How is a diagonalisable matrix $A$ expressed in terms of its eigenvalues and eigenvectors?

$A = PDP^{-1}$, where $D$ is diagonal containing the eigenvalues and the columns of $P$ are the corresponding eigenvectors. Then $A^{n} = PD^{n}P^{-1}$.

Define $\cosh x$, $\sinh x$ and $\tanh x$ in terms of exponentials.

$\cosh x = \dfrac{e^{x}+e^{-x}}{2}$, $\sinh x = \dfrac{e^{x}-e^{-x}}{2}$, $\tanh x = \dfrac{\sinh x}{\cosh x} = \dfrac{e^{x}-e^{-x}}{e^{x}+e^{-x}}$.

State the fundamental hyperbolic identity and the derivatives of $\sinh x$ and $\cosh x$.

Identity: $\cosh^{2}x - \sinh^{2}x = 1$. Derivatives: $\dfrac{d}{dx}\sinh x = \cosh x$ and $\dfrac{d}{dx}\cosh x = \sinh x$.

State Osborn's rule for converting trigonometric identities to hyperbolic ones, with an example.

Replace each trig function with its hyperbolic counterpart, but change the sign of any product (or implied product) of two sines. Example: $\cos 2x = 1 - 2\sin^{2}x$ becomes $\cosh 2x = 1 + 2\sinh^{2}x$.

Give the logarithmic form of the inverse hyperbolic function $\operatorname{arsinh} x$.

$\operatorname{arsinh} x = \ln\!\left(x + \sqrt{x^{2}+1}\right)$, valid for all real $x$.

What are the $n$th roots of unity, and how are they expressed?

The solutions of $z^{n}=1$: $z_k = e^{2\pi i k/n} = \cos\dfrac{2\pi k}{n} + i\sin\dfrac{2\pi k}{n}$ for $k = 0,1,\dots,n-1$. They lie equally spaced on the unit circle.

What is the sum of all $n$th roots of unity (for $n>1$), and what geometric/algebraic fact explains it?

The sum is $0$. The roots are the vertices of a regular $n$-gon centred at the origin, and they are the roots of $z^{n}-1=0$ whose coefficient of $z^{n-1}$ is zero (sum of roots $=0$).

How can the series $\sum_{r=0}^{n-1}\cos(r\theta)$ be summed using complex numbers?

Take the real part of the geometric series $\sum_{r=0}^{n-1} e^{ir\theta} = \dfrac{1-e^{in\theta}}{1-e^{i\theta}}$; $\sum\cos(r\theta)$ is its real part and $\sum\sin(r\theta)$ its imaginary part.

In polar coordinates, state the formula for the area enclosed by the curve $r = f(\theta)$ between $\theta = \alpha$ and $\theta = \beta$.

$A = \dfrac{1}{2}\displaystyle\int_{\alpha}^{\beta} r^{2}\,d\theta = \dfrac{1}{2}\int_{\alpha}^{\beta} \big(f(\theta)\big)^{2}\,d\theta$.