• understand what a matrix is;
• identify rows and columns;
• determine the size of a matrix;
• identify individual matrix entries;
• understand the notation $a_{ij}$;
• determine whether two matrices are equal.

Sometimes we need to organize many numbers at once.

Instead of writing them as a long list, we can arrange them into rows and columns. For example,

This rectangular arrangement of numbers is called a matrix.

For example,

contains two horizontal rows and three vertical columns:

• Row 1: $(1,\ 4,\ 7)$
• Row 2: $(2,\ 5,\ 8)$
• Column 1: $\begin{pmatrix}1\\2\end{pmatrix}$

If a matrix has $m$ rows and $n$ columns, its size is written as

So the order is always

For example,

has 2 rows and 3 columns. Therefore its size is

A very common mistake is to reverse the dimensions of a matrix. For example,

is not $3\times2$. It has 2 rows and 3 columns, so its size is

The individual numbers inside a matrix are called its entries or elements. Consider

Each number occupies a specific position, and we can name every entry using two indices. The general notation is

The first index tells us the row. The second index tells us the column. So $a_{ij}$ means:

Consider

The entry $a_{12}$ means row 1, column 2. The number in that position is $7$. Therefore,

Similarly, $a_{21}$ means row 2, column 1. Therefore,

Remember: $a_{ij}$ means

It does not mean column $i$, row $j$. For example,

Then $a_{12}=8$, while $a_{21}=3$. In general, and are different positions.

A matrix with $m$ rows and $n$ columns can be written as

We can write this more compactly as

Its size is $m\times n$. The entry in the final row and final column is $a_{mn}$.

Two matrices are equal when all corresponding entries are equal. For example,

and

Every entry of $A$ matches the entry in the same position in $B$. Therefore,

Now consider

and

Both matrices contain the same four numbers, but the entries are not in the same positions. For example, $a_{21}=4$ while $b_{21}=2$. Therefore,

Matrices $A=(a_{ij})$ and $B=(b_{ij})$ are equal if their corresponding entries are equal:

for every valid pair of indices $i,j$.

A matrix is a rectangular array of numbers arranged in rows and columns. A matrix with $m$ rows and $n$ columns has size

Matrix dimensions are always written as

Individual matrix entries are written as $a_{ij}$, where $i=\text{row number}$ and $j=\text{column number}$.

Two matrices are equal when all corresponding entries are equal: $A=B$ if $a_{ij}=b_{ij}$ for every corresponding position.

Great work! You now know how to:

• recognize a matrix;
• identify rows and columns;
• determine matrix dimensions;
• read entries using $a_{ij}$;
• compare two matrices for equality.