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Linear Algebra
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Page 2

Chapter-1 Mathematics

THE GATE ACADEMY PVT.LTD. H.O.: #74, Keshava Krupa (third Floor), 30
th

Cross, 10
th

Main, Jayanagar 4
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CHAPTER 1

Linear Algebra

Linear algebra comprises of the theory and applications of linear system of equations, linear
transformations and Eigen-value 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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Chapter-1 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

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 ) = [

]

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Chapter-1 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.

Gauss-Jordan 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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Chapter-1 Mathematics

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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”+