Matrix addition, subtraction, scalar multiplication and multiplication, the zero and identity matrices, non-commutativity, and the determinant of a 2x2 and 3x3 matrix as an area or volume scale factor.

What this dot point is asking

OCR wants you to handle the algebra of matrices fluently: add and subtract matrices of the same order, multiply a matrix by a scalar, multiply two matrices when their dimensions are compatible, recognise the zero and identity matrices and the fact that matrix multiplication is not commutative, and compute and interpret the determinant of a $2 \times 2$ and a $3 \times 3$ matrix as an area or volume scale factor.

Adding, subtracting and scaling

Two matrices can be added or subtracted only when they have the same order (same number of rows and columns), and you combine corresponding entries. Scalar multiplication multiplies every entry by the scalar. These operations obey the usual commutative and associative laws, so matrix addition behaves like ordinary addition.

Matrix multiplication

The product $\mathbf{A}\mathbf{B}$ is defined only when the number of columns of $\mathbf{A}$ equals the number of rows of $\mathbf{B}$. If $\mathbf{A}$ is $m \times n$ and $\mathbf{B}$ is $n \times p$, then $\mathbf{A}\mathbf{B}$ is $m \times p$, and the entry in row $i$, column $j$ is the sum of the products of row $i$ of $\mathbf{A}$ with column $j$ of $\mathbf{B}$.

Multiplication is associative, $(\mathbf{A}\mathbf{B})\mathbf{C} = \mathbf{A}(\mathbf{B}\mathbf{C})$, and distributes over addition, but it is not commutative: in general $\mathbf{A}\mathbf{B} \ne \mathbf{B}\mathbf{A}$, and indeed one product may be defined while the other is not. The identity matrix $\mathbf{I}$ (ones on the leading diagonal, zeros elsewhere) acts like the number $1$: $\mathbf{A}\mathbf{I} = \mathbf{I}\mathbf{A} = \mathbf{A}$.

The determinant of a 2x2 matrix

The determinant of a $2 \times 2$ matrix measures how the transformation it represents scales area. A determinant of zero means the transformation collapses the plane onto a line (the matrix is singular and has no inverse), and a negative determinant means the orientation is reversed (a reflection is involved).

The determinant of a 3x3 matrix

For a $3 \times 3$ matrix you expand along any row (or column) using cofactors: multiply each entry by the determinant of the $2 \times 2$ matrix left when its row and column are deleted, attaching the sign pattern $\begin{pmatrix} + & - & + \end{pmatrix}$ across the top row. The result is the volume scale factor of the transformation in three dimensions.

A useful property is the multiplicative rule $\det(\mathbf{A}\mathbf{B}) = \det\mathbf{A} \times \det\mathbf{B}$, which means a composite transformation scales area or volume by the product of the individual factors.

These operations underpin everything else in the matrices strand: inverses use the determinant, transformations use multiplication, and proof by induction often establishes a formula for a power of a matrix.

Try this

Q1. Find $\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} + 2\begin{pmatrix} 0 & 1 \\ -1 & 2 \end{pmatrix}$. [2 marks]

• Cue. $2\begin{pmatrix} 0 & 1 \\ -1 & 2 \end{pmatrix} = \begin{pmatrix} 0 & 2 \\ -2 & 4 \end{pmatrix}$, so the sum is $\begin{pmatrix} 1 & 4 \\ 1 & 8 \end{pmatrix}$.

Q2. Find the value of $k$ for which $\begin{pmatrix} k & 3 \\ 2 & k \end{pmatrix}$ is singular. [3 marks]

• Cue. Singular means $\det = 0$: $k^2 - 6 = 0$, so $k = \pm\sqrt{6}$.