Introduction to Matrices

A matrix (pronounced MAY-trix) is a rectangular arrangement of numbers, like: $$ \cssId{s2}{\begin{bmatrix} 7 & 3 & 2\\ 6 & 5 & 8 \end{bmatrix}} $$ A number in a matrix is often called an element, a member, or an entry of the matrix.
The members of a matrix are often enclosed in (square) brackets.

Rows have a horizontal orientation, and are numbered from top to bottom:

$7$	$3$	$2$

$6$	$5$	$8$

Thus, $ \cssId{s7}{\begin{bmatrix} 7 & 3 & 2\\ \end{bmatrix}} $ is the first row, and $ \cssId{s9}{\begin{bmatrix} 6 & 5 & 8\\ \end{bmatrix}} $ is the second row.

Columns have a vertical orientation, and are numbered from left to right:

$7$			$3$			$2$
$6$			$5$			$8$

Thus, $ \cssId{s13}{\begin{bmatrix} 7\\ 6 \end{bmatrix}} $ is the first column, $ \cssId{s15}{\begin{bmatrix} 3\\ 5 \end{bmatrix}} $ is the second column, and $ \cssId{s17}{\begin{bmatrix} 2\\ 8 \end{bmatrix}} $ is the third column.

The plural of matrix is matrices (pronounced MAY-tri-sees).

Matrices offer a way to represent large amounts of data in an organized way.
Matrices are particularly well-suited to computer analysis.
There are a multitude of applications of matrices, including:

encrypting numerical data (think national security)
computer graphics (think video games)
solving systems of equations

The size of a matrix is reported by stating the number of rows, followed by the ‘$\,\times\,$’ symbol, followed by the number of columns.
For example, the size of

$ \cssId{s30}{\begin{bmatrix} 7 & 3 & 2\\ 6 & 5 & 8 \end{bmatrix}} $

is $\,2\times 3\,$, which is read aloud as ‘$\,2\,$ by $\,3\,$’.

Observe that an $\,m \times n\,$ matrix has $\,mn\,$ elements,
since there are $\,m\,$ rows, with $\,n\,$ entries in each row.

Matrices are usually named with capital letters.
Members of a matrix are usually named with lowercase letters.
In particular, the elements of a matrix $\,M\,$ are conventionally named $\,m_{ij}\,$:

$\,m_{ij}\,$ is read aloud as ‘$\,m\,$ sub $\,i\,$ $\,j\,$’
the $\,i\,$ and $\,j\,$ are called subscripts; they are written a little bit below the line
the first subscript ($\,i\,$) gives the row number of the element
the second subscript ($\,j\,$) gives the column number of the element
a capital letter is used to denote the matrix, and the corresponding lowercase letter is used to denote the elements

For example, if

$ \cssId{s46}{M = \begin{bmatrix} 7 & 3 & 2\\ 6 & 5 & 8\end{bmatrix}} $

then:
$\,m_{11} = 7\,$ (first row, first column; read as em sub one one, NOT (say) em sub eleven)
$\,m_{12} = 3\,$ (first row, second column)
$\,m_{13} = 2\,$ (first row, third column)
$\,m_{21} = 6\,$ (second row, first column)
$\,m_{22} = 5\,$ (second row, second column)
$\,m_{23} = 8\,$ (second row, third column)

Two matrices are equal when they have the same size, and corresponding elements are equal.
Precisely, we have:

EQUALITY OF MATRICES

Let $\,A\,$ and $\,B\,$ be matrices.
Then,

$A=B$

if and only if

$A\,$ and $\,B\,$ have the same size
and
$\,a_{ij} = b_{ij}\,$ for all $\,i\,$ and $\,j\,$.

Here, $\,i\,$ takes on all possible row numbers, and $\,j\,$ takes on all possible column numbers.

A matrix with the same number of rows and columns is called a square matrix.
For example, $ \cssId{s76}{\begin{bmatrix} 1 & 2\\ 3 & 4 \end{bmatrix}} $ is a square matrix.

A matrix where all the entries are zero is called a zero matrix.
For example, $ \cssId{s80}{\begin{bmatrix} 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 \end{bmatrix}} $ is a $\,2\times 4\,$ zero matrix.

Master the ideas from this section
by practicing the exercise at the bottom of this page.

When you're done practicing, move on to:
Basic Arithmetic with Matrices

(MAX is 26; there are 26 different problem types.)