Matrix arithmeticAQA A-Level Further Maths: Revision notes

A matrix is a rectangular array of numbers (entries) in rows and columns. A matrix with $m$ rows and $n$ columns has order $m\times n$; a square matrix has $m=n$. Two matrices are equal only if they have the same order and every corresponding entry is equal, so a matrix equation gives one equation per entry. For example $\begin{pmatrix}1&2&2\\-1&3&1\end{pmatrix}$ has order $2\times3$.

Matrices of the same order are added or subtracted entry by entry. A matrix cannot be added to one of a different order. Multiplying by a scalar $k$ multiplies every entry by $k$. Example: with $\mathbf{C}=\begin{pmatrix}1&2\\3&4\end{pmatrix}$ and $\mathbf{D}=\begin{pmatrix}0&-1\\5&2\end{pmatrix}$, $2\mathbf{C}-\mathbf{D}=\begin{pmatrix}2-0&4+1\\6-5&8-2\end{pmatrix}=\begin{pmatrix}2&5\\1&6\end{pmatrix}$. Addition is commutative and associative, and $k(\mathbf{A}+\mathbf{B})=k\mathbf{A}+k\mathbf{B}$.

The product $\mathbf{AB}$ is defined only when the matrices are conformable: 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 $\mathbf{AB}$ is $m\times p$. Each entry is a row times column sum: entry $(i,j)$ of $\mathbf{AB}$ is row $i$ of $\mathbf{A}$ dotted with column $j$ of $\mathbf{B}$. Example: $\begin{pmatrix}1&0&2\\-1&3&1\end{pmatrix}\begin{pmatrix}2&1\\0&-1\\4&3\end{pmatrix}=\begin{pmatrix}1(2)+0(0)+2(4)&1(1)+0(-1)+2(3)\\-1(2)+3(0)+1(4)&-1(1)+3(-1)+1(3)\end{pmatrix}=\begin{pmatrix}10&7\\2&-1\end{pmatrix}$. Multiplication is associative, $(\mathbf{AB})\mathbf{C}=\mathbf{A}(\mathbf{BC})$, and distributive, $\mathbf{A}(\mathbf{B}+\mathbf{C})=\mathbf{AB}+\mathbf{AC}$, but it is not commutative: in general $\mathbf{AB}\ne\mathbf{BA}$ (and one product may be defined when the other is not).

The zero matrix $\mathbf{O}$ has every entry $0$: $\mathbf{A}+\mathbf{O}=\mathbf{A}$, $\mathbf{A}-\mathbf{A}=\mathbf{O}$ and $\mathbf{AO}=\mathbf{OA}=\mathbf{O}$ (when defined). The identity matrix $\mathbf{I}$ is square with $1$s on the leading diagonal and $0$s elsewhere, for example $\begin{pmatrix}1&0\\0&1\end{pmatrix}$. For a square matrix $\mathbf{A}$ of the same order, $\mathbf{AI}=\mathbf{IA}=\mathbf{A}$, so $\mathbf{I}$ plays the role of $1$. Unlike numbers, $\mathbf{AB}=\mathbf{O}$ does not mean $\mathbf{A}=\mathbf{O}$ or $\mathbf{B}=\mathbf{O}$.

The square of a square matrix is $\mathbf{M}^2=\mathbf{MM}$, and $\mathbf{M}^n$ is $n$ copies multiplied. Use $\mathbf{I}$ and scalar multiples to write relationships between powers. Example: for $\mathbf{M}=\begin{pmatrix}2&-1\\1&0\end{pmatrix}$, $\mathbf{M}^2=\begin{pmatrix}3&-2\\2&-1\end{pmatrix}=2\mathbf{M}-\mathbf{I}$. Then $\mathbf{M}^3=\mathbf{M}(2\mathbf{M}-\mathbf{I})=2\mathbf{M}^2-\mathbf{M}=3\mathbf{M}-2\mathbf{I}=\begin{pmatrix}4&-3\\3&-2\end{pmatrix}$. To find constants in an equation like $\mathbf{A}^2+p\mathbf{A}+q\mathbf{I}=\mathbf{O}$, work out each term, add, and equate every entry to $0$.