This document includes an in-depth exploration of matrices and systems of linear equations. It begins with an introduction to the basic concepts of matrices, covering matrix notation, types of matrices (such as square matrices, row matrices, and column matrices), and essential operations like matri...
, 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 - […….]
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
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
4
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