What is a Matrix?
This lesson introduces the matrix  the rectangular array at the heart of
matrix algebra. Matrix algebra is used quite a bit in advanced statistics, largely
because it provides two benefits.
 Efficient methods for manipulating sets of data and solving sets of
equations.
Matrix Definition
A matrix is a rectangular array of numbers arranged in
rows and columns. The array of numbers below is an example of a matrix.

21 
62 
33 
93 

44 
95 
66 
13 
77 
38 
79 
33 
The number of rows and columns that a matrix has is called its
dimension or its order. By convention,
rows are listed first; and columns, second. Thus, we would say that the
dimension (or order) of the above matrix is 3 x 4, meaning that it has
3 rows and 4 columns.
Numbers that appear in the rows and columns of a matrix are called
elements of the matrix. In the above matrix, the element
in the first column of the first row is 21; the element in the second
column of the first row is 62; and so on.
Matrix Notation
Statisticians use symbols to identify matrix elements and matrices.
Matrix elements. Consider the matrix below,
in which matrix elements are represented entirely by symbols.

A_{1}_{1} 
A_{1}_{2} 
A_{1}_{3} 
A_{1}_{4} 

A_{2}_{1} 
A_{2}_{2} 
A_{2}_{3} 
A_{2}_{4} 
By convention, first subscript refers to the
row number; and the second subscript, to the column number. Thus,
the first element in the first row is represented by
A_{1}_{1}. The second element in the first row is
represented by A_{1}_{2}. And so on,
until we reach the fourth element
in the second row, which is represented by
A_{2}_{4}.
Matrices. There are several ways to represent a
matrix symbolically. The simplest
is to use a boldface letter, such as A, B,
or C. Thus, A might represent a
2 x 4 matrix, as illustrated below.
Another approach for representing matrix A is:
A = [ A_{i}_{j} ] where i = 1, 2 and j = 1, 2, 3, 4
This notation indicates that A is a matrix with 2 rows and
4 columns. The actual elements of the array are not displayed; they are
represented by the symbol A_{i}_{j}.
Other matrix notation will be introduced as needed.
For a description of all the matrix notation used in this tutorial,
see the
Matrix Notation Appendix.
Matrix Equality
To understand matrix algebra, we need to understand matrix
equality. Two matrices are equal if all three of the following conditions
are met:
 Each matrix has the same number of rows.
 Each matrix has the same number of columns.
 Corresponding elements within each matrix are equal.
Consider the three matrices shown below.
If A = B, we know that x = 222 and
y = 333; since corresponding elements
of equal matrices are also equal. And we know that matrix C
is not equal to A or B, because
C has more columns than A or
B.
Test Your Understanding
Problem 1
The notation below describes two matrices  A and
B
A = [ A_{i}_{j} ] where i = 1, 2, 3 and j = 1, 2


B = 

111 
222 
333 
444 

555 
666 
777 
888 


Which of the following statements about A and
B are true?
I. Matrix A has 5 elements.
II. The dimension of matrix B is 4 x 2.
III. In matrix B, element B_{2}_{1} is
equal to 222.
(A) I only
(B) II only
(C) III only
(D) All of the above
(E) None of the above
Solution
The correct answer is (E).
 Matrix A has 3 rows and 2 columns; that is,
3 rows, each with 2 elements. This adds up to
6 elements, altogether  not 5.
 The dimension of matrix B
is 2 x 4  not 4 x 2. That is, matrix B has 2 rows
and 4 columns  not 4 rows and 2 columns.
 And, finally, element B_{2}_{1} refers to the
first element in the
second row of matrix B, which is equal to 555  not 222.