INTRODUCTION TO MATRICES

A matrix (pronounced MAY-trix) is a rectangular arrangement of numbers, like: [beautiful math coming... please be patient] $$ \begin{bmatrix} 7 & 3 & 2\\ 6 & 5 & 8 \end{bmatrix} $$ 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,   [beautiful math coming... please be patient] $ \begin{bmatrix} 7 & 3 & 2\\ \end{bmatrix} $   is the first row, and   [beautiful math coming... please be patient] $ \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,   [beautiful math coming... please be patient] $ \begin{bmatrix} 7\\ 6 \end{bmatrix} $   is the first column,   [beautiful math coming... please be patient] $ \begin{bmatrix} 3\\ 5 \end{bmatrix} $   is the second column, and   [beautiful math coming... please be patient] $ \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:

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

[beautiful math coming... please be patient] $ \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 [beautiful math coming... please be patient] $\,m_{ij}\,$:

For example, if

[beautiful math coming... please be patient] $ M = \begin{bmatrix} 7 & 3 & 2\\ 6 & 5 & 8\end{bmatrix} $

then:
[beautiful math coming... please be patient] $\,m_{11} = 7\,$   (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 [beautiful math coming... please be patient] $\,A\,$ and $\,B\,$ be matrices.
Then,
$A=B$ if and only if [beautiful math coming... please be patient] $A\,$ 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, [beautiful math coming... please be patient] $ \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, [beautiful math coming... please be patient] $ \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


On this exercise, you will not key in your answer.
However, you can check to see if your answer is correct.
PROBLEM TYPES:
1 2 3 4 5 6 7 8 9 10 11 12 13
14 15 16 17 18 19 20 21 22 23 24 25 26
AVAILABLE MASTERED IN PROGRESS

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