Department of Mathematics
UNIT: MATRICES AND
SYSTEMS OF LINEAR EQUATIONS
, MATRICES AND SYSTEMS OF LINEAR
EQUATIONS
After working through this chapter you should be able to:
• Identify different types of matrices.
• Perform detailed algebraic operations on matrices.
• Calculate determinants of square matrices.
• Solve systems of linear equations using Gaussian Elimination, Cramer’s Rule and the Inverse
matrix method.
• Write a matrix as a product of elementary matrices.
1 INTRODUCTION
The main objective of this chapter is the development of various techniques that can be employed to
solve a system of simultaneous equations such as :
2 x + 4 y + 6 z = 18
4 x + 5 y + 6 z = 24
3x + y − 2z = 4
You might recognize the notion of simultaneous equations from your school years or possibly from a
previous course in Algebra.
Before progressing to the techniques we first discuss the concept of a matrix and the determinant of a
matrix.
2 MATRICES
A matrix can be thought of as an ordered rectangular arrangement of numbers into horizontal
rows and vertical columns. To indicate that we are working with a matrix the numbers are enclosed
by square brackets - […….]
Examples of matrices:
1 −1 1 −1 0 1
2 −1 3 2 5 12 −4 6 , 4 , 0 −1 25 4 −16
1 6 0 , 4 15 , 0 −3 , [ ]
12 8 0 0 3 −2
DEFINITION
The size of a matrix is the number of rows by the number of columns.
The first matrix above has size “2 by 3”, writen as 2×3, as it has 2 rows and 3 columns. The number
of rows (horizontal) is always stated first and the number of columns (vertical) second. The sizes of
the other matrices above are: 3 × 2, 2 × 2, 3 × 3, 3 ×1 and 1× 5 respectively.
2
,Consider the system given in the introduction above.
The coefficients of x , y and z can be described by the matrix:
2 4 6
A = 4 5 6
3 1 −2
in which the coefficients of x appear in the first column, the coefficients of y in the second column
and the coefficients of z in the third column. This matrix is called the coefficient matrix of the
system. This matrix will be used later on in the solving of systems of linear equations.
Because A consists of three rows and three columns, we say A has size “3 by 3”, and write 3 × 3 for
the size of A .
At this point we formally define what is meant by the term matrix.
DEFINITION
Let m, n ∈ . An ordered rectangular array of real numbers into m rows and n colunms
a11 a12 a1n
...
a a22 ... a2 n
A = 21 = ( aij ) ; aij ∈
m×n
am1 am 2 ... amn
is called an m × n matrix over .
The aij appearing in the matrix are referred to as the entries of the matrix.
Specifically aij refers to the entry appearing in the i-th row and the j -th column of A.
Remarks:
• We use “matrix over ” to express the fact that the entries of the matrix are real numbers.
• Matrices are usually denoted with capital letters and the entries by small letters (lower case
letters).
Example 1
1 4 −6 8
Let A = 0 10 5 −2
14 9 0 3
Then A = ( aij ) with:
3× 4
a31 = 14
a22 = 10
a14 = 8
3
, DEFINITIONS
Two matrices A = ( aij ) and B = ( bij ) are equal if they have the same size
and the entries in corresponding positions are equal.
i.e. If A = ( aij ) and B = ( bij ) then A = B ⇔ aij = bij , i = 1, 2,..., m, j = 1, 2,..., n.
m× n m×n
If a matrix has the same number of rows as columns, it is called a square matrix.
A diagonal matrix is a square matrix A = ( aij ) where aij = 0 if i ≠ j.
m×m
a11 0 0
0 a22 0
0 0 amm
If each diagonal entry in a diagonal matrix is 1,
i.e. a11 = a22 = ... = amm = 1, then the matrix is called an identity matrix.
The n × n identity matrix is denoted by I n .
If A is an n × m matrix, the transpose of A, denoted by AT , is defined
to be the m × n matrix that results from interchanging the rows and columns from A.
The trace of a square n × n matrix A, denoted tr ( A ) , is the sum of the entries on the
diagonal of the matrix.
n
i.e. tr ( A ) = arr
r =1
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