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KIET GROUP OF INSTITUTIONS, DELHI NCR, GHAZIABADB. Tech. (I sem), 2020-21
MATHEMATICS-I (KAS-103T)
MODULE 1(MATRICES)
SYLLABUS: Types of Matrices: Symmetric, Skew-symmetric and Orthogonal Matrices;
Complex Matrices, Inverse and Rank of matrix using elementary transformations, Rank-Nullity
theorem; System of linear equations, Characteristic equation, Cayley-Hamilton Theorem and its
application, Eigen values and Eigen vectors; Diagonalisation of a Matrix.
CONTENTS
S.No MATRICES PAGE
NO.
1.1 Introduction 2
1.2 Types of Matrices 2
1.3 Basic Operations on Matrices 7
1.4 Transpose of a Matrix 9
1.5 Symmetric Matrix 10
1.6 Skew-Symmetric Matrix 10
1.7 Complex Matrices 11
1.8 Hermitian and Skew-Hermitian matrix 13
1.9 Unitary Matrix 13
1.10 Elementary Transformations (or Operations) 19
1.11 Inverse of a Matrix by E-Operations (Gauss-Jordan Method) 19
1.12 Rank of a Matrix 26
1.13 Nullity of a Matrix 27
1.14 Methods of Finding Rank of Matrix 28
1.15 Solution of System of Linear Equations 46
1.16 Linear Dependence and Independence of Vectors 60
1.17 Eigenvalues and Eigenvector 63
1.18 Cayley-Hamilton Theorem 75
1.19 Similarity Transformation 80
1.20 Diagonalization of a matrix 80
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, KIET GROUP OF INSTITUTIONS, DELHI NCR, GHAZIABAD
MATRICES
1.1 INTRODUCTION
The term ‘Matrix’ was given by J.J.Sylvester about1850, but was introduced first by Cayley in
1860. By a ‘matrix’ we mean “rectangular array” of numbers. Matrices (plural of matrix) find
applications in solution of system of linear equations, probability, mathematical economics,
quantum mechanics, electrical networks, curve fitting, transportation problems etc. Matrices are
easily agreeable for computers.
Matrix inverse can be used in Cryptography. It can provide a simple and effective procedure for
encoding and decoding messages.
Matrix
A rectangular array of m. n numbers (real or complex) arranged in m rows (horizontal lines) and
n columns (vertical lines) and enclosed in brackets [ ] is called a matrix of order m n .It is
also called m n matrix.
The numbers in the matrix are called entries or elements of the matrix.
Elements of a matrix are located by the double subscript ij where i denotes the row and j the
a11 a12 ... a1n
a a22 ... a2 n
21
column. The matrix A is written as A ... ... ... ...
am1 am 2 ... amn mn
If all these entries are real, then the matrix A is called a real matrix.
Note: Matrix has no numerical value.
1.2 TYPES OF MATRICES
(a) Square Matrix
In a matrix, if the number of rows = number of columns = n, then it is called a square matrix of
order n.
3 4 5
Example : A 5 6 7 is a square matrix since its order is 3 3.
9 3 2
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, KIET GROUP OF INSTITUTIONS, DELHI NCR, GHAZIABAD
(b) Row Matrix
A matrix with only one row is called a row matrix.
Examples :
Let A a11a12 a13 a14 .............a1n . It is a row matrix with n columns. So, it is a row
matrix of type 1 n .
Also, A 4 5 6 7 . It is a row matrix with 4 columns. So, it is a row matrix of type
1 4 .
(c) Column Matrix
A matrix with only one column is called a column matrix.
Examples :
a11
a
21
A a31
Let
.
an1
It is a column matrix with n rows. So, it is a column matrix of type n 1 .
1
2
Also, Let
A 3
4
5
It is a column matrix with 5 rows. So, it is a column matrix of type 5 1 .
(d) Diagonal Matrix
A square matrix in which all the non- diagonal elements are zero is called a diagonal matrix.
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, KIET GROUP OF INSTITUTIONS, DELHI NCR, GHAZIABAD
Example:
2 0 0
0 is a diagonal matrix of order 3.
Let A 0 3
0 0 4
(e) Scalar Matrix
In a diagonal matrix if all the diagonal elements are equal to a non-zero scalar , then it is called
a scalar matrix.
Example:
0 0
0 is a scalar matrix of order 3.
Let A 0
0 0
(f) Unit Matrix or Identity matrix
In a diagonal matrix if all the diagonal elements are equal to 1, then it is called a unit matrix or
identity matrix.
1 0 0
1 0
Examples : 1, 0
1
, 0 1
0 0 0 1
are identity matrices of orders 1, 2, 3 respectively.They are denoted by I1, I2, I3.
In general, In is the identity matrix of order n .
(g) Zero Matrix or Null matrix
In a matrix (rectangular or square), if all the entries are equal to zero, then it is called a zero
matrix or null matrix.
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