Online Learning Summit - Daffodil International University

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MATRIX-6

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OBJECTIVES

 Zero Row

 Non zero Row

 Pivot

 Elementary Row operation

 Echelon Matrix

 Row Echelon Form (REF)

 Row Reduced Echelon Form (RREF)

 Normal Form (NF)

 Rank

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( ⁻ ¹ ⁰ ⁰ ⁴ ² ³ ⁰ ⁰ ⁻ ⁴ ⁰ ⁰ ¹ ¹ ⁰ ⁰ ⁰ ^/ ² )

Zero Row

 Non Zero Row

 Pivot

Zero Row: All elements of the row are zero

Pivot: First non-zero element considered from the left.

Non-Zero Row: At least one element of the row is non-zero.

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Zero Row

 Non Zero Row

 Pivot

EXAMPLE

 







 









2 0

0 0

3 1

0 0

4 3

2 1 5 4 0 1

0 0 2 9

0 0 0 0

 

 

 

0 6 0 2 0

2 0 0 4 5

0 5 0 0 2

  

 

 

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Elementary Row operation a b c

m n p x y z

 

 

 

Row/Column

Exch ange ^R

^p

^ ^R

^q

Multiply or Divide by any Number

o r 1



_p



_p

k R R

k

Add or Subtract any two Row/Column

'

 

p p q

R R k R

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Row Echelon Form (REF):

1. Each Pivot of a row is in a

column to the right of the pivot of the above it.

2. All entries in a column below a pivot are zero.

3. All non-zero rows must lie above zero rows(if any).

REF

( ^{1 2} ^{0 3} ^{0 0} ^{0 0} ⁻ ⁴⁴ ⁴ ⁰ ¹ ¹ ⁰ ⁴ ⁰ ^/ ² )

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( ⁻ ¹ ⁰ ⁰ ⁴ ² ³ ⁰ ⁰ ⁻ ⁴ ⁰ ⁰ ¹ ¹ ⁰ ⁰ ⁰ ^/ ² )

Row Echelon Form

(REF):

1. Interchange positions of zero rows(whenever they appear) so that they lie below all nonzero rows.

2. Selection of pivots.

3. Formation of zeroes below pivots (using pivots ).

First Pivot Second Pivot

Third Pivot

( ¹ ⁰ ⁰ ⁰ ² ³ ⁸ ⁰ ⁻ ⁻ ⁴ ⁰ ¹ ⁴ ¹ ⁰ ⁰ ⁰ ^/ ² )

Multiplying and then adding with , we get

�

₃^′

= 4 �

₁

+ �

₃

( ¹ ⁰ ⁰ ⁰ ² ³ ⁰ ⁰ ⁻ ⁴⁴ ⁴ ⁰ ¹ ¹ ⁰ ⁴ ⁰ ^/ ² ) REF

Multiplying and and then adding together, we get

�

₃^{′ ′}

= 8 �

₂

− 3 �

₃^′

4 − 4 8 + 0 − 4 + 0 0 + 0

0 24 12 0

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Row Reduced Echelon Form (RREF):

1. Making zeroes above each pivot by means of that pivot.

2. Translate each pivot to 1

First Pivot Second Pivot

Third Pivot

( ¹ ⁰ ⁰ ⁰ ² ³ ⁰ ⁰ ⁻ ⁴ ¹ ⁰ ¹ ¹ ¹ ^/ ⁰ ⁰ ^/ ¹¹ ² )

Multiplying , we get

�

₃^′

= 1

44 �

₃

REF

( ¹ ⁰ ⁰ ⁰ ² ³ ⁰ ⁰ ⁻ ⁰ ¹ ⁰ ¹ ³ ¹ ^/ ^/ ⁰ ⁰ ²² ¹¹ )

Multiplying and then adding with , we get

�

₂^′

= �

₂

− 4 �

^′₃

( ¹ ⁰ ⁰ ⁰ ² ³ ⁰ ⁰ ⁻ ⁴⁴ ⁴ ⁰ ¹ ¹ ⁰ ⁴ ⁰ ^/ ² )

0 4 1

2 − 4 11

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( ¹ ⁰ ⁰ ⁰ ² ³ ⁰ ⁰ ⁰ ⁰ ¹ ⁰ ⁻ ³ ¹ ¹ ^/ ^/ ⁰ ²² ¹¹ ^/ ¹¹ )

�

₁^′

= �

₁

+ �

^′₃

0 +0 1+1 0 + 1

11 ( ¹ ⁰ ⁰ ⁰ ² ¹ ⁰ ⁰ ⁰ ⁰ ¹ ⁰ ¹ ¹ ¹ ^/ ^/ ^/ ⁰ ¹¹ ²² ¹¹ )

�

₂^{′ ′}

= 1

3 �

₂^′

( ¹ ⁰ ⁰ ⁰ ⁰ ¹ ⁰ ⁰ ⁰ ⁰ ¹ ⁰ ¹ ¹ ^/ ^/ ⁰ ⁰ ²² ¹¹ )

1 − 0 2 − 2 0 − 0 ₁₁ ¹ ⁻ ₁₁ ¹

�

₁^{′ ′}

= �

₁^′

− �

₂^{′ ′}

( ¹ ⁰ ⁰ ⁰ ⁰ ¹ ⁰ ⁰ ⁰ ⁰ ¹ ⁰ ¹ ¹ ^/ ^/ ⁰ ⁰ ²² ¹¹ )

RREF

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( ¹ ⁰ ⁰ ⁰ ⁰ ¹ ⁰ ⁰ ⁰ ⁰ ¹ ⁰ ¹ ¹ ^/ ^/ ⁰ ⁰ ²² ¹¹ )

�

₄^′

=�

₄

− 1 22 �

2

To find Normal Form of a matrix, we need to do column operation

( ¹ ⁰ ⁰ ⁰ ⁰ ¹ ⁰ ⁰ ⁰ ⁰ ¹ ⁰ ¹ ^/ ⁰ ⁰ ⁰ ¹¹ )

^�

⁴^′

^=�

⁴

⁻ ²² ¹ ( ^� ¹ ⁰ ⁰ ⁰

²

⁰ ¹ ⁰ ⁰ ⁰ ⁰ ¹ ⁰ ⁰ ⁰ ⁰ ⁰ )

( ¹ ⁰ ⁰ ⁰ ⁰ ¹ ⁰ ⁰ ⁰ ⁰ ¹ ⁰ ⁰ ⁰ ⁰ ⁰ )

( ^� ₀

³

⁰ ₀ )

NF

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Normal Form of a matrix

0 I

n

 

 

 

 I n  ⁰  

0 0 0

I n

 

 

 







I n

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Rank

The number of non-zero rows in REF is called rank

of the matrix.

( ¹ ⁰ ⁰ ⁰ ² ³ ⁰ ⁰ ⁻ ⁴⁴ ⁴ ⁰ ¹ ¹ ⁰ ⁴ ⁰ ^/ ² )

REF

The number of pivots in REF is called rank of the

matrix.

( ¹ ⁰ ⁰ ⁰ ² ³ ⁰ ⁰ ⁻ ⁴⁴ ⁴ ⁰ ¹ ¹ ⁰ ⁴ ⁰ ^/ ² )

REF

Rank

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~

RE F

(

¹⁰⁰ ⁻⁰⁰³² ⁰¹⁰ ⁻⁰⁰¹

) ^~ ( ¹ ⁰ ⁰ ⁰ ⁰ ⁰ ⁰ ¹ ⁰ ⁰ ⁰ ⁰ )

Example

RRE F

( ¹ ⁰ ⁰ ⁰ ¹ ⁰ ⁰ ⁰ ⁰ ⁰ ⁰ ⁰ )

~ ~ ( ^{⋯ ⋮ ⋯} ^� 0

²

^⋮ ⋮ ⁰ 0 ) ^NF

1 .   





 









0 1 6

4 4 2 6 4

2 1 3 2

 







 





 

4 1 0

0 0 0 0

0 2 1 3 2

~

2 .   





 









5 2

3 4 8

0 1 1

2 ~   





 









7 7

0 4 8

0 1 1

2 ~   





 









1 1

0 1 2

0 1 1

2  







 









1 0 0

1 2

0 1 1

~ 2

~

RE F

~

RRE F

NF

~ ^�

₄

Rank=

2 Rank=

4

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You can justify your understanding by answering the following questions

 What are the possible normal forms of matrices?

 How many EF, RREF and NF do exist for a matrix?

 When will you have RREF = NF ?

 What are the EF, RREF and NF of the following matrix ( 0 0 -2 3 0 4 1 0 ) ?

 What is the NF of I₅₁ ?

 Why can’t a matrix have 2 different normal forms ?

 What are possible normal forms of 2×3 and 2×2 matrices.

 Write a 5×4 matrix for which EF = RREF = NF .

 Which matrices do not posses any normal form ?

 What is the normal form of a zero matrix ?

 What is the rank of a 23 ×27 zero matrix ?

 Do the matrices & possess same rank ?

What is the rank of –233 I₁₃₇?

 What is the rank of a permutation matrix of order 27×27 ?

 What may be the maximum & minimum ranks of a 27×37 non-zero matrix ?

 M is 25×15 & rk(M) = 15. What will be its RREF & NF ?

 M is 4×8 & rk(M) = 3. What is it: True or False ?

 A rank deficient matrix has dimension 7×9. Comment on its rank.

 L is 129, M is 291; one of them is a null matrix. Find rk(ML).

 L is 129, M is 291; both are nonzero matrices. Find rk(LM).

S is a submatrix of M. Which is true among the followings ?

PRACTICE PERFECTION ⤑

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EXERCISE

Find Rank, RREF and NF for the followings:

1 .

2. 3. ( ⁴ ¹ ⁰ ⁻ ⁻ ⁻ ² ^{1 9 0} ⁷ ^{8 0 4} ^{5 0 2} ⁻ ⁵ )

4. 5 .

6 .

7 .

8 .

9 .

10.

Online Learning Summit - Daffodil International University

MATRIX-6

OBJECTIVES

 Zero Row

 Non zero Row

 Pivot

 Elementary Row operation

 Echelon Matrix

 Row Echelon Form (REF)

 Row Reduced Echelon Form (RREF)

 Normal Form (NF)

 Rank

( − 1 0 0 4 2 3 0 0 − 4 0 0 1 1 0 0 0 / 2 )

Zero Row

 Non Zero Row

 Pivot

Zero Row: All elements of the row are zero

Pivot: First non-zero element considered from the left.

Non-Zero Row: At least one element of the row is non-zero.

Zero Row

 Non Zero Row

 Pivot

EXAMPLE

 







 









2 0

0 0

3 1

0 0

4 3

2

1 5 4 0 1

0 0 2 9

0 0 0 0

 

 

 

 

 

0 6 0 2 0

2 0 0 4 5

0 5 0 0 2

  

 

 

 

 

Elementary Row operation a b c

m n p x y z

 

 

 

 

 

Row/Column

Exch ange R

 R

Multiply or Divide by any Number

o r 1





k R R

k

Add or Subtract any two Row/Column

 

R R k R

Row Echelon Form (REF):

1. Each Pivot of a row is in a

column to the right of the pivot of the above it.

2. All entries in a column below a pivot are zero.

3. All non-zero rows must lie above zero rows(if any).

REF

( 1 2 0 3 0 0 0 0 − 44 4 0 1 1 0 4 0 / 2 )

( ⁻ ¹ ⁰ ⁰ ⁴ ² ³ ⁰ ⁰ ⁻ ⁴ ⁰ ⁰ ¹ ¹ ⁰ ⁰ ⁰ ^/ ² )

Exch ange ^R

^ ^R

( ^{1 2} ^{0 3} ^{0 0} ^{0 0} ⁻ ⁴⁴ ⁴ ⁰ ¹ ¹ ⁰ ⁴ ⁰ ^/ ² )

( ⁻ ¹ ⁰ ⁰ ⁴ ² ³ ⁰ ⁰ ⁻ ⁴ ⁰ ⁰ ¹ ¹ ⁰ ⁰ ⁰ ^/ ² )

( ¹ ⁰ ⁰ ⁰ ² ³ ⁸ ⁰ ⁻ ⁻ ⁴ ⁰ ¹ ⁴ ¹ ⁰ ⁰ ⁰ ^/ ² )

( ¹ ⁰ ⁰ ⁰ ² ³ ⁰ ⁰ ⁻ ⁴⁴ ⁴ ⁰ ¹ ¹ ⁰ ⁴ ⁰ ^/ ² ) REF

( ¹ ⁰ ⁰ ⁰ ² ³ ⁰ ⁰ ⁻ ⁴ ¹ ⁰ ¹ ¹ ¹ ^/ ⁰ ⁰ ^/ ¹¹ ² )

( ¹ ⁰ ⁰ ⁰ ² ³ ⁰ ⁰ ⁻ ⁰ ¹ ⁰ ¹ ³ ¹ ^/ ^/ ⁰ ⁰ ²² ¹¹ )

( ¹ ⁰ ⁰ ⁰ ² ³ ⁰ ⁰ ⁻ ⁴⁴ ⁴ ⁰ ¹ ¹ ⁰ ⁴ ⁰ ^/ ² )

( ¹ ⁰ ⁰ ⁰ ² ³ ⁰ ⁰ ⁰ ⁰ ¹ ⁰ ⁻ ³ ¹ ¹ ^/ ^/ ⁰ ²² ¹¹ ^/ ¹¹ )

( ¹ ⁰ ⁰ ⁰ ² ¹ ⁰ ⁰ ⁰ ⁰ ¹ ⁰ ¹ ¹ ¹ ^/ ^/ ^/ ⁰ ¹¹ ²² ¹¹ )