What Is a Matrix and What Do the Terms Element, Row, Column, and Order Mean?

A matrix is a rectangular array of numbers or other mathematical objects arranged in rows and columns. Its order states its number of rows followed by its number of columns.

Matrix Terminology

Consider the matrix

$$A=\begin{pmatrix} 4 & -2 & 7 \\ 1 & 5 & 0 \end{pmatrix}.$$

An element is identified using subscript notation. If the element in row $i$ and column $j$ is written as $a_{ij}$, the row number always comes first.

For the example above:

• $a_{12}=-2$, because $-2$ is in row $1$, column $2$.
• $$, because is in row , column .

In general, a matrix with $m$ rows and $n$ columns has order $m\times n$ and contains (mn) elements. Therefore, this $2\times 3$ matrix contains $2(3)=6$ elements.

The order also determines which matrix operations are possible. Two matrices can be added or subtracted only when they have the same order, because corresponding elements must pair. For multiplication, the number of columns in the first matrix must equal the number of rows in the second matrix. Thus, a $2\times 3$ matrix can multiply a $3\times 4$ matrix, producing a $2\times 4$ matrix, but the reverse product is not automatically defined.

A common misconception is to write columns before rows when giving the order. Always read matrix dimensions as rows by columns, not columns by rows.

Exam Technique

This is AHL 1.14, Introduction to matrices, so it is HL-only content in IB Mathematics: Applications and Interpretation. In an exam, identify the order before performing matrix operations, and interpret $a_{ij}$ as “row $i$, column $j$.” This becomes essential when checking whether matrix addition or multiplication is defined.