Chapter-2 - Lecture notes 2Linear Systems:The linear system is a combination of 2 or more linear equation (Mathisfun, n.d) where those equations are sharing same points, In graphically when you draw those points on the graph, it will draw straight lines w PDF

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Download Chapter-2 - Lecture notes 2Linear Systems:The linear system is a combination of 2 or more linear equation (Mathisfun, n.d) where those equations are sharing same points, In graphically when you draw those points on the graph, it will draw straight lines w PDF

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Chapter 2 Matrices, Determinants and Systems of Linear Equations 2.1. Definition of matrix and basic operations N . A rectangular array of numbers in

Definition: Let m and n be in A=

(

R

)

a11 a12 ⋯ a1 n ⋮ ⋱ ⋮ a m 1 a m 2 ⋯ a mn

is called a matrix in

R.

Remark: The numbers in the matrix are called the entries of the matrix. Note: 1. A is an m× n matrix if A has m rows(horizontals) and n columns(verticals). 2. aij is the element that appears in the ith row and in the jth column. a (¿¿ ij)m ×n is an m× n matrix. 3. A=¿ 4. m × n is called the size of the matrix. Example 1: Consider A=

(−12

3 8 7 11

)

. The size of this matrix is 2 ×3 .

a11 =−1, a22=7 and a13=8 a (¿¿ ij)m ×n and A=¿ corresponding elements are equal.

b (¿¿ ij)m ×n are equal, written as A = B, iff their B=¿

Definition: Two matrices

Example 2: Consider the two matrices given below A=

( −12

)

(

3 8 −1 4 8 andB= 7 11 2 7 11

)

Since a12=3 ≠ 4=b 12 , we can say that A ≠ B . Definition: A 1 ×n matrix is called a row vector (row matrix) and an m × 1matrix is called a column vector (column matrix).

1

Example 3: Example 4:

1 4 7

()

is a 3 × 1 column matrix.

( 3 −9 0 ) is a1 × 3 row matrix.

Example 5: (8) is both a column and row matrix. Definition: A matrix in which the number of rows and columns are equal i.e., an called a square matrix of order n. Example 6:

(

2 0 4 A= 3 −9 5 1 −3 8

)

n ×n is

is a square matrix of order 3.

Operation with matrices Addition (Subtraction) of Matrices. a Definition: Let (¿¿ ij)m ×n A=¿

Then

a (¿ ¿ ij+b ij )m ×n A + B =¿

and

b and (¿¿ ij)m ×n be two matrices. B=¿ a (¿ ¿ ij−bij )m × n A−B=¿

Definition 3.1.2 If A and B are matrices with the same size, then we define the sum A + B to be the matrix obtained by adding the entries of B to the corresponding entries of A, and we define the difference A - B to be the matrix obtained by subtracting the entries of B from the corresponding entries of A. If A = [aij ] and B = [bij] have the same size, then this definition states that (A + B)ij = (A)ij + (B)ij = aij + bij (A - B)ij = (A)ij - (B)ij = aij - bij Example 7.

2

Note: For

a (¿¿ ij)m × n and A=¿  

b (¿¿ ij)m × n and B=¿

c (¿¿ ij)m ×n C=¿

A + B=B+ A A + ( B+C )= ( A+ B) +C

Definition: A matrix all of whose elements are zero is called a zero matrix and it is denoted by O . Example 8:

(

O 2 ×3= 0 0 0 0 0 0

Definition: Let

a (¿¿ ij)m ×n A=¿

)

be a matrix and ∝∈ R , the scalar multiple of A by ∝ is

∝a (¿¿ ij)m ×n . ∝ A=¿

Example: If

(

2 0 4 A= 3 −9 5 1 −3 8

)

, then

(

)

8 0 16 4 A= 12 −36 20 4 −12 32

2.2. Product of Matrices and some algebraic properties; transpose of matrix a Let (¿¿ ij)m ×n and A=¿

b (¿¿ ij)n × p . We define the product B=¿

AB=(C ik )m× p Example 10:

Example 11: Let

A=

(−14 11)∧B=(52 −31 ) 3

AB

by

Then Compute A. AB B. BA C. Is AB =BA ? Properties of Matrix Multiplication i) ii) iii) iv)

AB ≠ BA A ( BC )=( AB)C A ( B+ C )= AB+ AC∧ ( A +B ) C= AC + BC OA=O = AO

Definition: A square matrix in which all but the diagonal elements are zero is called diagonal matrix.

Definition:A diagonal matrix whose all of its diagonal elements are equal is called Scalar matrix.

Definition: A scalar matrix whose all of its diagonal elements are one is called the Identity matrix.

a Definition: Let (¿¿ ij)m ×n be a matrix. We define the transpose of A denoted by A=¿ be the = n ×m matrix where the ijt h entry is a ji . 4

A

t

to

Example 13:

Properties of Matrix Transpose Let A, B be matrices over i. ii. iii.

( At )t = A ( A+ B)t= A t + Bt

iv.

( AB)t =Bt A t

t

(∝ A) =∝ A

R

and

∝∈ R

t

Special Matrices 1. Symmetric Matrix is a matrix which is equal to its transpose. i.e. Example 14:

(

A= A

t

)

1 −1 4 t A= 1 2 6 = A 4 6 3

2. Skew Symmetric matrix is a matrix which is equal to -1times its transpose. i.e. A=−A t . Example 14: Theorem: Let i. ii.

(

)

0 −1 6 t A= 1 0 1 =− A −6 −1 0

A be a square matrix. Then

A + At is symetric. A − A t is skew symmetric. a 3. A square matrix (¿¿ ij)n ×n is said to be an upper triangular matrix if A=¿ aij=0, ∀i> j and strictly upper triangular matrix if aij=0, ∀i≥ j . a A square matrix (¿¿ ij)n ×n is said to be an upper triangular matrix if A=¿ aij=0, ∀i< j and strictly upper triangular matrix if aij=0, ∀i≤ j .

5

2.3. Elementary Row Operations and Echelon Form Let A be an m× n matrix. The elementary row operation on A is A j ,i (Interchanging two rows) i. A i →∝ A i (Multiplication of a row by a non-zero constant ∝ ) ii. A i → Ai +∝ A j (Addition of a constant multiple of one row to another row) iii. Definition: Two matrices are equivalent written as A the other by a sequence of elementary row operations. Example:

A 2,3

Thus

(

(

) (

1 4 −1 1 4 −1 4 A2 3 6 0 12 24 0 1 5 8 1 5 8

) )(

B if one can be obtained from

)

(

)

1 4 −1 −35 −68 −1 A1 → A 1−3 A 3 1 5 8 1 5 8 12 24 0 12 24 0

(

1 4 −1 3 6 0 1 5 8

−35 −68 −1 1 5 8 12 24 0

) Row Echelon Form

Definition: A matrix is in a row echelon form if it satisfies the following conditions 1. Any row (if any) consisting of entirely of zeros appears at the bottom of the matrix. 2. The first non-zero number in any row not consisting of entirely zero is 1 (leading 1). 3. If two successive rows do not consist entirely of zeros, the leading 1 in the lower row occurs farther to the right than the leading 1 in the higher row. Example: The following matrices are in row echelon form 6

0 0 0 1 2 42 1 0 55 A= 0 0 0 B= 0 1 0 3 ∧C= 0 1 0 0 0 0 0 0 0 14 0 0 00

(

) (

) (

)

The following matrices are not in row echelon form

(

) (

) (

)

1 2 6 1 2 58 1 3 40 D= 0 0 0 E= 0 0 1 7 ∧F= 0 1 9 8 0 1 0 0 0 05 0 1 97

Definition: A matrix in a row echelon form is said to be in reduced row echelon form if all entries in any column containing the leading 1is zero. Example: From the above matrices, matrix A and C are in reduced row echelon form. 2.4. Rank of a matrix Definition: Let A be an m × n matrix. Let A R be the row echelon form of A. The rank R( A) is the number of non-zero rows of the row echelon form of A ( A R ). R ( A )≤min (m , n) Example: Find the rank of the following matrices.

(

) ( ) (

)

1 0 0 1 2 1 3 40 A= 0 1 0 B= 2 4 C= 0 1 9 8 0 0 1 0 1 97 −3 −6 Answer:

ρ ( A )=¿ 3,

ρ (B )=1 2.5. Determinant of a Matrix and its Properties

a (¿¿ ij)n ×n is associated a number or an expression called the A=¿ determinant of A which is denoted by det (A) or |A| .

To every square matrix

Determinant of order one Let A=( a11) . Then det (A) ¿ a11 . Determinant of order two Let A=

(

)

a11 a12 a 21 a22

. Then det

( A )=a11 a22−a21 a 12

7

Determinant of order n n

det ( A ) =∑ (−1)i + j aij det ( Ai , j ) , 1≤ i≤ n j=1

expansion along the i

th

row and the

i th row where column. j

is the matrix formed by deleting (crossout) the

Ai, j

th

Example: Find the determinants of the following matrices

(

1 2 0 A= 3 6 1 1 −1 5

1.

)

(

)

3 4 5 B= 0 −2 8 11 2 6

2.

Solution: 1. Here n=3 and we can choose

i =1, i =2∨i =3

choosing i=1

n

A=¿ ∑ (−1 )i+ j a ij det ( A i , j ) ,1 ≤i ≤ n j=1

det ¿ 1+1

¿(−1 )

1+2

a11 det A 1,1 +(−1 )

1 +3

a12 det A1,2 +(−1 )

a13 det A 1,3

¿ a11 det A 1,1−a12 det A 1,2 +a13 det A1,3

¿ 1 ( 31) −2 (14 ) −4 ( 0 ) =3 Properties of Determinants 1. If two columns are interchanged, then the determinant changes by sign. 1 8 4 8 1 4 Example: 3 −1 5 = −1 3 5 2 2 7 2 2 7 2. If one column is a scalar multiple of the other column, then the determinant is zero.

|

||

| |

| |

|

1 8 2 Example: 3 −1 6 =0 2 2 4 3. If one of the columns is zero, then the determinant is zero. 0 8 4 Example: 0 −1 5 =0 0 2 7 4. If a scalar multiple of one column is added to another column, then the determinant doesn’t change. 8

Example:

1 8 4 1 8 6 3 −1 5 = 3 −1 11 2 2 7 2 2 11

| | |

|| | |

|

5. The determinant of a diagonal matrix is the product of elements in the diagonal. 1 0 0 Example: 0 −1 0 =1 ×−1× 7=−7 0 0 7 6. The determinant of an upper triangular matrix is the product of elements in the diagonal. 1 8 4 Example: 0 −1 5 =1×−1 × 7 =−7 0 0 7 7. 8.

det A =det A t det ( AB ) =det ( A ) . det(B) 1 −1 9. det ( A )= det ( A) 10. Suppose that A 1 , A 2 , … , A j , … , A n are columns of matrix A. 1 2 j k n 1 2 k n 1 2 j n det ( A , A , … , A + A , … , A ) =det ( A , A , … , A , … , A ) + det (A , A , … , A , … , A )

Example: 1 2 1 1 4 1 1 2+4 1 det 3 5+4 9 =det 3 5 9 +de t 3 4 9 −9 6 8 −9 9 8 −9 6+ 9 8

(

) (

11. Suppose that

) (

)

A 1 , A 2 , … , A j , … , A n are columns of matrix A.

det ( A 1 , A2 , … , n A j , … , A n) =n det ( A1 , A 2 , … , n A j , …, A n) 2.6. Inverse of Matrix and its Properties Definition: A matrix A is said to be non singular or invertible if there exists a unique matrix I n . We say B is the multiplicative inverse of A . The B such that AB =BA =¿ unique inverse B of A is denoted by A−1 . If such a matrix doesn’t exist, we say matrix A is singular or non-invertible. Note: Inverse of a matrix is only defined for square matrices. Moreover, not all square matrices are invertible. Properties of Invertible matrices Let A i.

and B t −1

be invertible matrices of order n . Then −1 t

( A ) =( A )

9

( A−1 )−1 = A

ii. iii.

( AB)−1=B−1 A−1

Example: Show that 1.

( 10 −14 )

2.

(

is the inverse of itself.

1 2 3 2 5 7 −2 −4 −5

)

is the inverse of

(

3 −4 2

−2 −1 1 −1 0 1

)

.

Gauss-Jordan elimination for computing inverse of a matrix n with matrix

of order

n to form the

i.

Adjoin the identity matrix of order new matrix ( A|I n ) .

ii.

Compute the reduced row echelon form of matrix ( A|I n ) . If this reduced row echelon form is of type (I n|B ) , then B is the inverse of A . If the reduced row echelon form is not of type ( I n|B ) , in that the matrix to the right is not I n , then A is singular.

A

Example: Find the inverse of the following matrices. Solution:

(

( 1 2|1 0 ) R → R −2 R (10 01|10 −21 )

( A|I n ) = 0 1 0 1

1

1

2

)

−1 ∴ A = 1 −2 0 1

ADJOINT OF A MATRIX Definition: Let M ij be a matrix which obtained by omitting the i th row and the j th column of an n ×n square matrix A . det M ij is called the minor of the element aij of the matrix A . Let A ij=(−1)i + j det M ij . Then A ij

is called the cofactor of the element aij of the matrix A . Then the transpose of the matrix found from A ij that is ( Aij)t is called adjoint of A which is denoted by adj( A) .

Example: Let

(

)

−1 3 6 A= 6 5 2 −3 −3 −6

then we have the following:

10

|−35 −62 |=−30+6=−24. 5 2 a (¿−1)is (−1 ) | =−24= A . −3 −6|

 The minor of a11 (¿−1)

is

1 +1

 The cofactor of

11

11

Similarly A 12 =30, A 13 =−3, A 21 =0, A 22=24, A 23 =−12 , A 31 =−24, A 32 =38 A 33 =−23.

(

Then the matrix found from the cofactors is

, AdjA

2.7.

is

(

)

) )

−24 30 −3 0 24 −12 −24 38 −23

(

t

. Therefore the adjoint of

−24 30 −3 −24 0 −24 24 38 . 0 24 −12 i .e AdjA= 30 −24 38 −23 −3 −12 −23

Systems of Linear Equations

Definition: Equations of the form a11 x 1+ a12 x 2 +...+a1 n x n=b1

a21 x 1 +a 22 x 2+...+ a2 n x n= b2 ⋮

am 1 x 1+ am 2 x 2 +...+ amn x n=b m are called systems of

m linear equations in

n unknowns.

The above system of linear equation can be written as AX =b , where

(

)

a11 a12 ⋯ a1 n A= ⋮ ⋱ ⋮ a m 1 a m 2 ⋯ a mn Note: the entries of

Here

A

k

th

() ()

x1 b1 x b , X = 2 ∧b= 2 ⋮ ⋮ xn bm column of

A are the coefficients of the variable

is called the coefficient matrix. If

()

0 b= 0 ⋮ 0 11

x k in X .

, the system is called homogenous;

A

otherwise it is called non-homogenous. A homogenous system has at least one solution, the trivial solution i.e. x 1=x 2=…= xn =0 . The m ×( n+1) matrix whose 1st n columns are the columns of A and whose last column is b which is called the augmented matrix denoted by (A│b). Example: Consider the following systems of linear equations

{

2 x 1+ 4 x 2−x 3=1 x 1+ 5 x 3=0 9 x1 +7 x 2−x 3=5

Here

(

2 4 −1 A= 1 0 5 9 7 −1

)

and

()()

1 0 b= 0 ≠ 0 5 0

. Hence the system is non-homogenous. Here

is the coefficient matrix and the augmented matrix is

(

|)

2 4 −1 1 1 0 5 0 9 7 −1 5

A

.

Note:  

When a system of linear equation has solution, then it is called consistent otherwise it is known as inconsistent. Two systems of linear equations are said to be equivalent if they have the same solution.

Example:

=3 ∧ =3 {2x−x + yy=6 {−2 xx−+5yy=−6

are equivalent.

Methods of Solving Systems of Linear Equations 1. Gaussian Elimination Method Let AX =b

be a system of linear equations, then

i.

Write down the augmented matrix of the system of linear equations in the form of B=( A|b ) .

ii.

Apply the elementary row operations on the matrix B . A. If rank of A=¿ Rank of B=n , the system has a solution. ¿ n , the system has infinitely many solutions. B. If Rank A=¿ Rank B C. If Rank A ¿ Rank B , the system has no solution. Use back substitution.

iii.

Example: Using Gaussian elimination method, solve the following systems of linear equations.

12

x 1−2 x 2 +3 x 3=9 −x 1+3 x 2=−4 2 x 1−5 x 2 + 5 x 3 =17

1.

Solution:

ii.

1 −2 3 9 B=( A|b )= −1 3 0 −4 2 −5 5 17

iii.

1 −2 3 9 R2 → R2 + R1 1 −2 3 9 3 5 −1 3 0 −4 R3 → R3 −2 R1 0 1 0 −1 −1 −1 2 −5 5 17

3.

{ {

|)

(

Since Rank

iv.

|)

(

R3 → R3 + R2

2.

) () ( )

(

i.

x1 9 1 −2 3 AX =b , where A= −1 3 0 , X= x 2 ∧b= −4 2 −5 5 17 x3

(

(

|)

|) ( |)

1 −2 3 9 R → 1 R 1 −2 3 9 0 1 35 3 2 3 0 1 3 5 0 0 24 0 0 12

B=3=¿ Rank A=¿ order of A=n , the system has exactly one solution.

Using back substitution x 3=2 x 2+3 x 3=5 ⇒x 2=−1 x 1−2 x 2 +3 x 2=9 ⇒x 1=1 x+ 4 z=1 x + y +10 z=10 2 x− y +2 z=−5 x + y −3 z =−1 y− z=0 −x +2 y=1

2. Cramer’s Rule a). If a linear system of equations consisting of n equations with the same number of unknowns, x 1 , x 2 , … , x n given by a11 x 1+ a12 x 2 +...+a1 n x n=b1

a21 x 1 +a 22 x 2+...+ a2 n x n= b2 13

⋮ am 1 x 1+ am 2 x 2 +...+ amn x n=b m

has a non-zero coefficient determinant D = det A, the system has precisely one solution. This solution is given by the formula x 1=

D D D1 , x 2= 2 , … x n= n , D D D

where D k is the determinant obtained from D by replacing the column with the entries b1 , b2 , … ,b n .

k th column in D by the

b). Hence if the system is homogenous and D ≠ 0 , it has only the trivial solution x 1= x 2=…= xn =0. If D=0 , the homogenous system also has non-trivial solutions. Example: Solve each of the following systems of linear equations using Cramer’s rule.

{

i. 2 x−5 y=23 4 x +6 y=−2 Solution:

A=

(24

−5 6

ii.

{

5 x + y−z =4 9 x+ y−z=1 x− y +5 z=2

)

23 −5 2 23 | | −2 6 | 4 −2 | x= =4 y= =−3

|A|

| A|

s . s .={ ( 4, −3 ) }

Thus

3. Inverse Method −1 −1 AX =B , where A isinvertible , A ( AX )= A B

Given

−1

⇒X= A Bis the unique solution.

Theorem: An n ×n matrix A is invertible if and only if

detA ≠ 0.

Example: Solve the following systems of linear equations using inverse method. a).

{

x −2 y =1 4 x− y =−2

b).

{

x −2 y +2 z=3 2 x + y +z=0 x+ z=−2

14

Solution: The system of linear equation can be expressed in its equivalent matrix form as AX =B , where A= −1 7 Since A = −4 7 −1

1 , X =( x )∧B=( ) ( 14 −2 ) −1 −2 y

( ) 2 7 1 7

,

−1 X =A B= 7 −4 7 −1

2 −5 7 1 7 = −6 1 −2 7 7

( )( ) ( )

Exercise

4. Solve each of the following equations using

EIGEN VALUES AND EIGEN VECTURES Matrix eigen value problems concern on the solution of vector equations AX =λX … … … … … … … .1

Where A is a given square matrix, X is unknown vector and λ is unknown scalar. 15

Clearly, X =0 is a solution of equation (1 ) , giving 0=0. but this has no application, thus we want to find solution vectors X ≠ 0 of equation (1) called Eigenvectors (characteristic vectors) of A. A .

How to find an Eigen value of AX =λX ⇔AX − λX =0

⇔(...