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Chapter 4

Matrices and Determinants

T his chapter is an introduction to matrices and matrix operations. Determinants, Cramer’s

rule, and Gauss’s elimination method are introduced. Some definitions and examples are
not applicable to subsequent material presented in this text, but are included for subject
continuity, and reference to more advance topics in matrix theory. These are denoted with a dag-
ger ( † ) and may be skipped.

4.1 Matrix Definition

A matrix is a rectangular array of numbers such as those shown below.

1 3 1
23 7 or
–
1 – 1 5 – 2 15
4 – 7 6
In general form, a matrix is denoted as
A

a 11 a 12 a 13 … a 1n
a 21 a 22 a 23 … a 2n
A = a 31 a 32 a 33 … a 3n (4.1)

… … ………
a m1 a m2 a m3 … a mn

j
i
The numbers a ij are the elements of the matrix where the index indicates the row, and indi-
cates the column in which each element is positioned. Thus, a 43 indicates the element posi-
tioned in the fourth row and third column.

A matrix of rows and columns is said to be of m × n order matrix.
n
m
If m = n , the matrix is said to be a square matrix of order (or ). Thus, if a matrix has five rows
m
n
and five columns, it is said to be a square matrix of order 5.

In a square matrix, the elements a , 11 a , 22 a , 33 … a nn are called the main diagonal elements.
,
,
Alternately, we say that the matrix elements a , 11 a , 22 a , 33 … a nn , are located on the main
diagonal.

Numerical Analysis Using MATLAB® and Excel®, Third Edition 4−1
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