1.Algebra Booster

CHAPTER
7 Matrices and Determinants
INTRODUCTION
1. Matrix
CONCEPT BOOSTER
A set of mn numbers (real or complex) arranged in the form
of a rectangular array having m rows and n columns is called
an m ¥ n matrix. We read as m by n matrix.
An m ¥ n matrix is usually written as
Èa11 a12 … a1n
˘
Í
a21 a22 … a
˙
Í
2n ˙
Í … … … … ˙
Í
˙
Îa
m1 am2
… amn˚
It is also denoted as [ a ] ¥ .
ij m
Note A matrix is not a number. It just an ordered collection
of numbers arranged in the form of a rectangular array.
1.1 Order
If a matrix has m rows and n columns, the order of the matrix
is m by n or m ¥ n.
(i) The order of the matrix [1 2 3] is 1 ¥ 3.
È1 2 3˘
(ii) The order of the matrix
Í
4 3 2
˙
Í ˙
is 3 ¥ 3.
ÍÎ5 6 4˙˚
È1 4 8˘
(iii) The order of the matrix Í
2 3 5
˙ is 2 ¥ 3.
Î ˚
2. TYPES OF MATRICES
(i) Row matrix
A matrix having only one row is called a row matrix.
For example, Let A = [1 2 3].
n
(ii) Column matrix
A matrix having only one column is called a column matrix.
For example,
È1˘
Í
2
˙
B = Í ˙.
Í 5 ˙
Í ˙
Î9˚
(iii) Rectangular matrix
A matrix in which number of rows and number of columns
are not equal is called a rectangular matrix.
È1 4 8˘
For example, C = Í
2 3 5
˙.
Î ˚
(iv) Square matrix
In a matrix, in which the number of rows is equal to the number
of columns, it is called a square matrix.
For example,
Ê1 2 3ˆ
Èa
b˘
A = Í
c d ˙
Î ˚ and B = Á2 3 4˜
Á ˜
Ë8 7 6¯
are square matrices of order 2 and 3 respectively.
(v) Diagonal matrix
In a square matrix, if all the diagonal elements are non-zero
and rest are zero is called a diagonal matrix.
For example,
1 0 È2 0 0˘
È ˘
A= , B
Í 0 4 0 ˙
Í ,
0 5
˙ =
Í ˙
etc.
Î ˚
ÍÎ0 0 7˙˚
(vi) Scalar matrix
In a square matrix, if all the diagonal elements are the same
and rest of the elements are zero, it is called a scalar matrix.
2 0 0
For example,
4 0 È ˘
È ˘
A= , B
Í 0 2 0 ˙
Í
0 4
˙ =
Í ˙
Î ˚
ÍÎ0 0 2˙˚