MatricesIB MYP Maths Extended: Revision notes

A matrix is a rectangular array of numbers called entries, written in brackets. Its order is $m\times n$: $m$ rows and then $n$ columns. For example $\begin{pmatrix}1&2&3\\ 4&0&1\end{pmatrix}$ has 2 rows and 3 columns, so its order is $2\times3$. A matrix with the same number of rows and columns is square. A matrix with one column, such as $\begin{pmatrix}2\\ 5\\ 12\end{pmatrix}$, has order $3\times1$. Two matrices are equal only if they have the same order and equal entries.

You can only add or subtract matrices of the same order. Do it entry by entry. To multiply by a number (a scalar), multiply every entry. Worked example. With $\mathbf A=\begin{pmatrix}2&-1\\ 0&3\end{pmatrix}$ and $\mathbf B=\begin{pmatrix}1&4\\ -2&5\end{pmatrix}$: $2\mathbf A-\mathbf B=\begin{pmatrix}4&-2\\ 0&6\end{pmatrix}-\begin{pmatrix}1&4\\ -2&5\end{pmatrix}=\begin{pmatrix}3&-6\\ 2&1\end{pmatrix}.$ Adding is commutative ($\mathbf A+\mathbf B=\mathbf B+\mathbf A$).

The product $\mathbf{AB}$ exists only if 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 of $\mathbf{AB}$ is a row of $\mathbf A$ times a column of $\mathbf B$: multiply matching entries and add. Worked example. $\mathbf M=\begin{pmatrix}1&2&3\\ 4&0&1\end{pmatrix}$ is $2\times3$ and $\mathbf N=\begin{pmatrix}2&1\\ 0&3\\ 1&-1\end{pmatrix}$ is $3\times2$, so $\mathbf{MN}$ is $2\times2$. The entry in row 2, column 1 is $4(2)+0(0)+1(1)=9$. In full, $\mathbf{MN}=\begin{pmatrix}5&4\\ 9&3\end{pmatrix}$. Matrix multiplication is not commutative: $\mathbf{AB}$ and $\mathbf{BA}$ are usually different. For $\mathbf A=\begin{pmatrix}2&-1\\ 0&3\end{pmatrix}$ and $\mathbf B=\begin{pmatrix}1&4\\ -2&5\end{pmatrix}$, $\mathbf{AB}=\begin{pmatrix}4&3\\ -6&15\end{pmatrix}$ but $\mathbf{BA}=\begin{pmatrix}2&11\\ -4&17\end{pmatrix}$.

Matrix multiplication organises totals. If $\mathbf S$ shows items sold (rows: shops, columns: products) and $\mathbf P$ is a column of prices, then $\mathbf{SP}$ gives the takings of each shop. Worked example. $\mathbf S=\begin{pmatrix}120&80&40\\ 90&100&60\end{pmatrix}$ and $\mathbf P=\begin{pmatrix}2\\ 5\\ 12\end{pmatrix}$ gives $\mathbf{SP}=\begin{pmatrix}1120\\ 1400\end{pmatrix}$: Dubai takes 1120 AED and Sharjah 1400 AED. A matrix scaled by 1.2 models a 20% rise in every entry. Always say what the rows and columns mean, and what assumptions the model makes.

A network is made of points (vertices) joined by lines (edges). Its adjacency matrix has one row and column for each vertex. The entry is the number of direct routes between the two vertices, so 1 if joined and 0 if not. For a two-way network the matrix is symmetrical. Worked example. Towns $W, X, Y, Z$ with routes $WX$, $WY$, $XY$, $YZ$ give $\mathbf M=\begin{pmatrix}0&1&1&0\\ 1&0&1&0\\ 1&1&0&1\\ 0&0&1&0\end{pmatrix}.$

The entries of $\mathbf M^n$ give the number of different journeys using exactly $n$ routes. For $\mathbf M^2$, each entry is a row of $\mathbf M$ times a column of $\mathbf M$. $\mathbf M^2=\begin{pmatrix}2&1&1&1\\ 1&2&1&1\\ 1&1&3&0\\ 1&1&0&1\end{pmatrix}$ The entry from $W$ to $Z$ is 1: the only two-route journey is $W\to Y\to Z$. The entry from $Y$ to $Y$ is 3: $Y\to W\to Y$, $Y\to X\to Y$ and $Y\to Z\to Y$. For three routes, work out $\mathbf M^3=\mathbf M\mathbf M^2$.