Class notes
Matrix for Engineering Mathes
- Course
- Institution
This is good learning notes for Engineering mathes.
[Show more]
Seller
Follow
mohd2125csit1129
Content preview
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
1
, 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
2
, 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.
3
, 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.
4
The benefits of buying summaries with Stuvia:
Guaranteed quality through customer reviews
Stuvia customers have reviewed more than 700,000 summaries. This how you know that you are buying the best documents.
Quick and easy check-out
You can quickly pay through credit card or Stuvia-credit for the summaries. There is no membership needed.
Focus on what matters
Your fellow students write the study notes themselves, which is why the documents are always reliable and up-to-date. This ensures you quickly get to the core!
Frequently asked questions
What do I get when I buy this document?
You get a PDF, available immediately after your purchase. The purchased document is accessible anytime, anywhere and indefinitely through your profile.
Satisfaction guarantee: how does it work?
Our satisfaction guarantee ensures that you always find a study document that suits you well. You fill out a form, and our customer service team takes care of the rest.
Who am I buying these notes from?
Stuvia is a marketplace, so you are not buying this document from us, but from seller mohd2125csit1129. Stuvia facilitates payment to the seller.
Will I be stuck with a subscription?
No, you only buy these notes for $77.75. You're not tied to anything after your purchase.
Can Stuvia be trusted?
4.6 stars on Google & Trustpilot (+1000 reviews)
56326 documents were sold in the last 30 days
Founded in 2010, the go-to place to buy study notes for 14 years now