IB Maths AI 1.11 Notes

Fundamentals of matrices

Matrices provide a compact way to organise numbers and perform calculations involving several quantities at once. They are especially useful for solving systems of equations, representing transformations and modelling practical situations.

A matrix is a rectangular array of numbers arranged in rows and columns.

For example:

$A=\begin{pmatrix}1&&2&&3\\4&&5&&6\end{pmatrix}$

This matrix has $2$ rows and $3$ columns, so its order is $2\times3$.

Each number in a matrix is called an element. The element in row $i$ and column $j$ is written as $a_{ij}$.

For the matrix $A=\begin{pmatrix}1&&2&&3\\4&&5&&6\end{pmatrix}$, $a_{12}=2$ because it is in the first row and second column, and $a_{23}=6$ because it is in the second row and third column.

Two matrices are equal only if they have the same order and all corresponding elements are equal.

For example, $\begin{pmatrix}1&&2\\3&&4\end{pmatrix}=\begin{pmatrix}1&&2\\3&&4\end{pmatrix}$ but $\begin{pmatrix}1&&2\\3&&4\end{pmatrix}\ne\begin{pmatrix}1&&2\\4&&3\end{pmatrix}$.

Matrices can be used to organise and solve real-world problems. They are useful whenever several related quantities must be handled together. They can be used for coding and decoding messages, for modelling movement between states, and for solving systems of equations efficiently.