A matrix is simply a grid of numbers, arranged in rows and columns. Think of a spreadsheet, a calendar, or even a tic-tac-toe board. These are all ways of organizing information in a rectangular layout. In mathematics, we use matrices to work with collections of data in a structured way.

We write matrices by enclosing the grid of numbers in brackets. The horizontal lines of numbers are called rows, and the vertical lines are called columns.

In this example, the first row is 1 2 3 and the second row is 4 5 6. The first column is 1 4, the second is 2 5, and the third is 3 6.

The size of a matrix is described by its dimensions: the number of rows and columns it has. We write this as "rows by columns." The matrix above has 2 rows and 3 columns, so its dimensions are $2 \times 3$.

To talk about a specific number within a matrix, we use subscripts. If we name our matrix $A$, the element in the $i$-th row and $j$-th column is written as $A_{ij}$. The row number always comes first.

A general matrix with $m$ rows and $n$ columns can be written like this:

Matrices come in a few common shapes and sizes. The names are straightforward and describe their appearance.

This is a $1 \times 4$ row matrix.

This is a $3 \times 1$ column matrix.

This is a $2 \times 2$ square matrix. A tic-tac-toe board could be represented by a $3 \times 3$ square matrix.

Let's review these new terms.

Ready to check your understanding?

Understanding these basic definitions is the first step. Next, we'll see how to perform calculations with matrices.