Introduction to Matrices

Definition

A $m\ast n$ $matrix$ is a rectangle of numbers having $m$ rows by $n$ columns, enclosed by brackets. For example:

$\left(\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\end{array}\right)$

is a 2x3 matrix and

$\left(\begin{array}{cc}1& 2\\ 4& 5\\ 7& 8\end{array}\right)$

is a 3x2 matrix.

Notes

Please note these matrices are not the “same size” - they do have the same amount of values inside, but they must have the same amount of rows and the same amount of columns to be considered the same size.

A matrix that only has one row is (sometimes) called a row vector, and a matrix with only one column is called a column vector.

Matrices are normally labelled using capital letters like $A,B,C$ etc.

If a matrix has the same number of rows as colums (e.g a 3x3 matrix), it is called a square matrix.

Values are denoted like so: $a_{bc}$

The number appearing in the $B$th row and the $C$th column is called the ($b$,$c$) entry of $A$.

As an example: If we take $A$ to be a $2\ast 3$ matrix like so: $\left(\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\end{array}\right)$

Then $a_{21}$ = 4, as 4 is in the 2nd column and the 1st row.

And as a final note: Two matrices are equal only if they have the same size and the values inside are all equal.