Matrix arithmeticEdexcel International A Level Further Maths: Revision notes

A matrix is a rectangular array of numbers in rows and columns. A matrix with $m$ rows and $n$ columns has order $m\times n$. Here we work mainly with $2\times2$ matrices and $2\times1$ column vectors. Each number is an entry (or element). Two matrices are equal only if they have the same order and every pair of corresponding entries is equal. This is used to find unknown entries: if $\begin{pmatrix}a+6&2+2b\\ 1&3\end{pmatrix}=\begin{pmatrix}10&8\\ 1&3\end{pmatrix}$ then $a+6=10$ and $2+2b=8$, so $a=4$ and $b=3$.

Matrices of the same order are added or subtracted entry by entry: $\begin{pmatrix}a&b\\ c&d\end{pmatrix}+\begin{pmatrix}e&f\\ g&h\end{pmatrix}=\begin{pmatrix}a+e&b+f\\ c+g&d+h\end{pmatrix}.$ Addition is commutative ($\mathbf A+\mathbf B=\mathbf B+\mathbf A$) and associative. Matrices of different orders cannot be added. Example: $\begin{pmatrix}2&-1\\ 3&4\end{pmatrix}+\begin{pmatrix}1&0\\ -2&5\end{pmatrix}=\begin{pmatrix}3&-1\\ 1&9\end{pmatrix}$.

A scalar is an ordinary number. Multiplying a matrix by a scalar $k$ multiplies every entry by $k$: $k\begin{pmatrix}a&b\\ c&d\end{pmatrix}=\begin{pmatrix}ka&kb\\ kc&kd\end{pmatrix}.$ Combine with addition and subtraction in the usual order of operations: $3\mathbf A-\mathbf B$ means multiply $\mathbf A$ by 3, then subtract $\mathbf B$. Example: $3\begin{pmatrix}2&-1\\ 3&4\end{pmatrix}-\begin{pmatrix}1&0\\ -2&5\end{pmatrix}=\begin{pmatrix}6&-3\\ 9&12\end{pmatrix}-\begin{pmatrix}1&0\\ -2&5\end{pmatrix}=\begin{pmatrix}5&-3\\ 11&7\end{pmatrix}$.

The product $\mathbf{AB}$ exists only 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. Each entry of $\mathbf{AB}$ is a row of $\mathbf A$ times a column of $\mathbf B$: multiply corresponding entries and add. $\begin{pmatrix}a&b\\ c&d\end{pmatrix}\begin{pmatrix}e&f\\ g&h\end{pmatrix}=\begin{pmatrix}ae+bg&af+bh\\ ce+dg&cf+dh\end{pmatrix}.$ Example: $\begin{pmatrix}2&-1\\ 3&4\end{pmatrix}\begin{pmatrix}1&0\\ -2&5\end{pmatrix}=\begin{pmatrix}4&-5\\ -5&20\end{pmatrix}$. A matrix times a column vector works the same way: $\begin{pmatrix}1&2\\ 3&4\end{pmatrix}\begin{pmatrix}5\\6\end{pmatrix}=\begin{pmatrix}17\\39\end{pmatrix}$.

• Not commutative: in general $\mathbf{AB}\neq\mathbf{BA}$. The order of multiplication matters, so always keep the order given.
• Associative: $(\mathbf{AB})\mathbf C=\mathbf A(\mathbf{BC})$.
• Distributive: $\mathbf A(\mathbf B+\mathbf C)=\mathbf{AB}+\mathbf{AC}$.
• Identity: $\mathbf I=\begin{pmatrix}1&0\\ 0&1\end{pmatrix}$ satisfies $\mathbf{AI}=\mathbf{IA}=\mathbf A$.
• Powers: $\mathbf A^2=\mathbf{AA}$. Entries are not squared: for $\mathbf M=\begin{pmatrix}1&2\\ 0&-1\end{pmatrix}$, $\mathbf M^2=\mathbf I$, not $\begin{pmatrix}1&4\\ 0&1\end{pmatrix}$. Context example: if $\mathbf S$ holds drinks sold per branch and $\begin{pmatrix}2\\3\end{pmatrix}$ the prices, then $\mathbf S\begin{pmatrix}2\\3\end{pmatrix}$ gives each branch's takings.