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Chapter1 Mathematics
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CHAPTER 1
Linear Algebra
Linear algebra comprises of the theory and applications of linear system of equations, linear
transformations and Eigenvalue problems.
Matrix
Definition
A system of “m n” numbers arranged along m rows and n columns
Conventionally, single capital letter is used to denote matrices
Thus,
A = [
a a a
a a a
a
a
a
a
a a a
]
a ith row, jth column
Types of Matrices
1. Row and Column matrices
Row Matrices [ 2, 7, 8, 9] single row ( or row vector)
Column Matrices [
] single column (or column vector)
2. Square matrix
Same number of rows and columns.
Order of Square matrix no. of rows or columns
e.g. A = [
] ; order of this matrix is 3
Principal Diagonal (or main diagonal or leading diagonal)
The diagonal of a square matrix (from the top left to the bottom right) is called as
principal diagonal.
Trace of the Matrix
The sum of the diagonal elements of a square matrix.
 tr (λ A) = λ tr(A) , λ is scalar
 tr ( A+B) = tr (A) + tr (B)
 tr (AB) = tr (BA)
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Chapter1 Mathematics
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Example
Demonstrate by example that AB BA
Solution
Suppose, A = 0
1, B= 0
1
AB = 0
1 , BA = 0
1
Example
Demonstrate by example that AB = 0 A = 0 or B = 0 or BA = 0
Solution
AB = 0
1 . 0
1 = 0
1
BA = 0
1
Example
Demonstrate that AC = AD C = D (even when A 0)
Solution
AC = 0
1 . 0
1 = 0
1
AD = 0
1 . 0
1 = 0
1
Although AC = AD, but C D
Example
Write the following matrix A as a sum of symmetric and skew symmetric matrix
A = [
]
Solution
Symmetric matrix =
(A +A ) =
{[
] [
]}
=
[
] = [
]
Skew symmetric matrix =
(A A ) = [
]
Page 11
Chapter1 Mathematics
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Example
Check, if the following matrix A is orthogonal.
A =
[
]
Solution
A = A =
[
]
A . A = A . A =
[
] = [
] = ,
Hence A is orthogonal matrix
Elementary transformation of matrix
1. Interchange of any 2 lines
2. Multiplication of a line by a constant (e.g. k )
3. Addition of constant multiplication of any line to the another line (e. g. + p )
Note
Elementary transformations don’t change the rank of the matrix.
However it changes the Eigen value of the matrix.
We call a linear system S1 “ ow Equivalent” to linear system S2, if S1 can be obtained from
S2 by finite number of elementary row operations.
GaussJordan method of finding Inverse
Elementary row transformations which reduces a given square matrix A to the unit matrix, when
applied to unit matrix , gives the inverse of A
Example
Find the inverse of [
]
Solution
Write in the form, [
]
Operate + , 
[
]
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=
giving the Eigen vector (4, 1)
For λ = 1, 0
1 0
x
x
1 = 0
4x + 4x = 0 ,
x + x = 0
=
giving the Eigen vector (1, 1)
Cayley Hamilton Theorem
Every square matrix satisfies its own characteristic equation.
Linear Dependence of Vectors
Vector: Any quantity having n components is called a vector of order n.
If one vector can be written as linear combination of others, the vector is linearly
dependent.
Linearly Independent Vectors
If no vectors can be written as a linear combination of others, then they are linearly
independent.
Suppose the vectors are x , x , x , x
Its linear combination is λ x + λ x + λ x + λ x = 0
If λ , λ , λ , λ are not “all zero” they are linearly dependent.
If all λ are zero they are linearly independent.
Example
If A = [
] , find the value of A A
Solution
A   = 0 λ λ λ
= (λ λ ) (λ+1) = 0
λ λ =0 or λ = 0
From Cayley Hamilton Theorem, A = 0 or A+I=0
As A
Hence, A = 0
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Example
For matrix A =[
], find Eigen vector of 3A A A .
Solution
A = 0

 = 0
(1 ) (3  ) (2 ) = 0 = 1, = 3, = 2
Eigen value of A = 1, 3, 2
Eigen value of A = 1, 27, 8
Eigen value of A = 1, 9, 4
Eigen value of = 1, 1, 1
First Eigen value of 3A A A = 3 = 4
Second Eigen value of 3A A A = 3 (27)+5(9)–6(3)+2=81+45–18+2 = 110
Third Eigen value of 3A A A =3(8)+5(4)–6(2)+2= 24 + 20 + 12 + 2 = 10
Example
Find Eigen values of matrix A = [
a
a
a
a
a
a
a
a
a
a
]
Solution
A  = 
a
a
a
a
a
a
a
a
a
a
 = 0
Expanding, (a ) (a ) (a ) (a ) = 0
= a , a , a , a which are just the diagonal elements
{Note: recall the property of Eigen value, “The Eigen value of triangular matrix are just the
diagonal elements of the matrix”+
Chapter1 Mathematics
THE GATE ACADEMY PVT.LTD. H.O.: #74, Keshava Krupa (third Floor), 30
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th
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CHAPTER 1
Linear Algebra
Linear algebra comprises of the theory and applications of linear system of equations, linear
transformations and Eigenvalue problems.
Matrix
Definition
A system of “m n” numbers arranged along m rows and n columns
Conventionally, single capital letter is used to denote matrices
Thus,
A = [
a a a
a a a
a
a
a
a
a a a
]
a ith row, jth column
Types of Matrices
1. Row and Column matrices
Row Matrices [ 2, 7, 8, 9] single row ( or row vector)
Column Matrices [
] single column (or column vector)
2. Square matrix
Same number of rows and columns.
Order of Square matrix no. of rows or columns
e.g. A = [
] ; order of this matrix is 3
Principal Diagonal (or main diagonal or leading diagonal)
The diagonal of a square matrix (from the top left to the bottom right) is called as
principal diagonal.
Trace of the Matrix
The sum of the diagonal elements of a square matrix.
 tr (λ A) = λ tr(A) , λ is scalar
 tr ( A+B) = tr (A) + tr (B)
 tr (AB) = tr (BA)
Page 10
Chapter1 Mathematics
THE GATE ACADEMY PVT.LTD. H.O.: #74, Keshava Krupa (third Floor), 30
th
Cross, 10
th
Main, Jayanagar 4
th
Block, Bangalore11
: 08065700750, [email protected] © Copyright reserved. Web: www.thegateacademy.com Page 9
Example
Demonstrate by example that AB BA
Solution
Suppose, A = 0
1, B= 0
1
AB = 0
1 , BA = 0
1
Example
Demonstrate by example that AB = 0 A = 0 or B = 0 or BA = 0
Solution
AB = 0
1 . 0
1 = 0
1
BA = 0
1
Example
Demonstrate that AC = AD C = D (even when A 0)
Solution
AC = 0
1 . 0
1 = 0
1
AD = 0
1 . 0
1 = 0
1
Although AC = AD, but C D
Example
Write the following matrix A as a sum of symmetric and skew symmetric matrix
A = [
]
Solution
Symmetric matrix =
(A +A ) =
{[
] [
]}
=
[
] = [
]
Skew symmetric matrix =
(A A ) = [
]
Page 11
Chapter1 Mathematics
THE GATE ACADEMY PVT.LTD. H.O.: #74, Keshava Krupa (third Floor), 30
th
Cross, 10
th
Main, Jayanagar 4
th
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Example
Check, if the following matrix A is orthogonal.
A =
[
]
Solution
A = A =
[
]
A . A = A . A =
[
] = [
] = ,
Hence A is orthogonal matrix
Elementary transformation of matrix
1. Interchange of any 2 lines
2. Multiplication of a line by a constant (e.g. k )
3. Addition of constant multiplication of any line to the another line (e. g. + p )
Note
Elementary transformations don’t change the rank of the matrix.
However it changes the Eigen value of the matrix.
We call a linear system S1 “ ow Equivalent” to linear system S2, if S1 can be obtained from
S2 by finite number of elementary row operations.
GaussJordan method of finding Inverse
Elementary row transformations which reduces a given square matrix A to the unit matrix, when
applied to unit matrix , gives the inverse of A
Example
Find the inverse of [
]
Solution
Write in the form, [
]
Operate + , 
[
]
Page 20
Chapter1 Mathematics
THE GATE ACADEMY PVT.LTD. H.O.: #74, Keshava Krupa (third Floor), 30
th
Cross, 10
th
Main, Jayanagar 4
th
Block, Bangalore11
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=
giving the Eigen vector (4, 1)
For λ = 1, 0
1 0
x
x
1 = 0
4x + 4x = 0 ,
x + x = 0
=
giving the Eigen vector (1, 1)
Cayley Hamilton Theorem
Every square matrix satisfies its own characteristic equation.
Linear Dependence of Vectors
Vector: Any quantity having n components is called a vector of order n.
If one vector can be written as linear combination of others, the vector is linearly
dependent.
Linearly Independent Vectors
If no vectors can be written as a linear combination of others, then they are linearly
independent.
Suppose the vectors are x , x , x , x
Its linear combination is λ x + λ x + λ x + λ x = 0
If λ , λ , λ , λ are not “all zero” they are linearly dependent.
If all λ are zero they are linearly independent.
Example
If A = [
] , find the value of A A
Solution
A   = 0 λ λ λ
= (λ λ ) (λ+1) = 0
λ λ =0 or λ = 0
From Cayley Hamilton Theorem, A = 0 or A+I=0
As A
Hence, A = 0
Page 21
Chapter1 Mathematics
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th
Cross, 10
th
Main, Jayanagar 4
th
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Example
For matrix A =[
], find Eigen vector of 3A A A .
Solution
A = 0

 = 0
(1 ) (3  ) (2 ) = 0 = 1, = 3, = 2
Eigen value of A = 1, 3, 2
Eigen value of A = 1, 27, 8
Eigen value of A = 1, 9, 4
Eigen value of = 1, 1, 1
First Eigen value of 3A A A = 3 = 4
Second Eigen value of 3A A A = 3 (27)+5(9)–6(3)+2=81+45–18+2 = 110
Third Eigen value of 3A A A =3(8)+5(4)–6(2)+2= 24 + 20 + 12 + 2 = 10
Example
Find Eigen values of matrix A = [
a
a
a
a
a
a
a
a
a
a
]
Solution
A  = 
a
a
a
a
a
a
a
a
a
a
 = 0
Expanding, (a ) (a ) (a ) (a ) = 0
= a , a , a , a which are just the diagonal elements
{Note: recall the property of Eigen value, “The Eigen value of triangular matrix are just the
diagonal elements of the matrix”+