This document provides an overview of topics in linear algebra and differential equations including: - Types of matrices such as row, column, square, diagonal, and identity matrices - Matrix operations including transpose, rank, and elementary row transformations - Converting matrices to row echelon form and using this to find the matrix rank - Solving systems of linear equations and determining the possible number of solutions depending on the values of parameters in the system.

1. Linear Algebra and Differential Equations
Unit I
Rank of Matrix, System of linear equations, Vector space,
Subspace of vector space, Linear span, Linear independence and
dependence, Basis, Dimension.
Duration: 10 Hrs.

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8. Types of Matrices
• Row and Column Matrix
• Square Matrix
• Diagonal Matrix
• Scalar Matrix
• Identity /Unit Matrix
• Zero Matrix
• Transpose of a Matrix
• Symmetric and Skew Symmetric Matrix
• Orthogonal matrix.
• Hermitian Matrix
• Skew Hermitian Matrix
• Unitary Matrix
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9. R A N K O F A M A T R I X
Let A is a nonzero matrix. Then the integer
r is called the rank of A if:
1.There exists at least one minor of order r
of A which is non zero and
2.Every minor of order greater than r is
zero.
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• Find the rank of matrix











3
8
4
2
6
3
1
4
2
A
2
0
2
2
6
1
4
0
)
24
24
(
1
)
8
9
(
4
)
16
18
(
2
|
|











RankA
Further
A
• Solution

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Remarks……
 The rank of a null matrix is zero.
 The rank of a nonsingular square matrix of order r is r.
 The rank of matrix remains unchanged by elementary transformations.
 The rank of transpose of matrix is equal to rank of matrix.
 The rank of a unit matrix of order n is n.
 The rank of diagonal matrix of order n whose all diagonal elements are
non zero is n.
 The rank of product of two matrices cannot exceed the rank of either
matrix.
 Rank of any matrix ≥1.

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E L E M E N A T R Y T R A N S F O R M A T I O N
Following changes made in the elements of any matrix are called
elementary transactions:
(i) Interchanging any two rows (or columns) .
(ii) Multiplying all the elements of any row (or column) by a non-zero
real number.
(iii) Adding non-zero scalar multitudes of all the elements of any row
(or columns) into the corresponding elements of any another row (or
column).

13. E C H E L O N F O R M
A rectangular matrix is in echelon form if it has the following properties:
• All nonzero rows are above any rows of all zeros
• Each leading entry of a row is in a column to the right of the leading entry of
the row above it
• All entries in a column below a leading entry are zeros
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H o w t o Tr a n s f o r m a M a t r i x I n t o I t s E c h e l o n
F o r m s & t o f i n d r a n k o f M a t r i x ?
By a series of finite number of row transformations, a matrix can be
transformed into an row echelon form which is not unique.
1. If 𝒂𝟏𝟏 is zero, use row operation 𝑹𝒊𝒋 and bring the non zero element in the
first row.
2. Now if 𝒂𝟏𝟏 ≠ 𝟏 then use row operation and convert it into unity.
3. Using suitable row operations, convert 𝒂𝟐𝟏 𝒂𝟑𝟏 &……into zero.
4. Repeat this procedure for other rows.
5. Rank of matrix is number of non-zero rows.

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Find the rank of the matrix (by minor form)
P R O B L E M S
2 1 1
0 3 2
2 4 3
A

 
 
 
 
 

 
1 2 3
2 4 6
3 6 9
A
 
 
  
 
 

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P R O B L E M S
Reduce the matrix to row-echelon form and find its rank
1 2 1
2 1 0
3 3 1
4 5 2
 
 
 
 
 
 
1 1 2 6
3 7 4 8
2 8 1 9

 
 

 
 

 
1 2 1 3
4 1 2 1
3 1 1 2
1 2 0 1

 
 
 
 

 
 

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P R O B L E M S
Analyse the rank of the matrix for different values of k
1 3
4 3
2 1 2
k
k
 
 
 
 
 

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P R O B L E M S
For what values of p the matrix
has (i) rank 1, (ii) rank 2, or (iii) rank 3.
2
2
2
p p
A p p
p p
 
 
  
 
 

20. System of linear equations
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21. P R O B L E M S
System of linear equations
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22. P R O B L E M S
System of linear equations
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23. P R O B L E M S
System of linear equations
P R O B L E M S
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24. P R O B L E M S
System of linear equations
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26. P R O B L E M S
3 3 2 1;
2 4;
10 3 2
2 3 5
x y z
x y
y z
x y z
  
 
  
  
Solve:
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P R O B L E M S
2 1;
2 9;
2 2
x y z
x y z
x y z
  
  
  
2; 1; 3
x y z
  

28. P R O B L E M S
5 3 7 4;
3 26 2 9;
7 2 10 5
x y z
x y z
x y z
  
  
  
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29. P R O B L E M S
System of linear equations
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30. P R O B L E M S
Solve:
3 2 0;
2 4 0;
11 14 0
x y z
x y z
x y z
  
  
  
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P R O B L E M S
2 0;
3 2 4 0;
4 3 3 0
x y z
x y z
x y z
  
  
  

32. P R O B L E M S
4. For what value of k the equations are consistent? Also solve
the system for these values of k.
Determine value of a and b for which the system has (i) no
solution (ii) unique solution (iii) infinite number of solutions.
Find the solution in case (ii) and (iii).
2
2 3;
;
3 3
x y z
x y z k
x y z k
  
  
  
5.
6;
2 3 10;
2
x y z
x y z
x y az b
  
  
  
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33. P R O B L E M S
6. For what value of λ the equations will have no unique
solutions? Will the equations have any solution for this value
of λ?
3 2 1;
2 2;
2 1
x y z
x y z
x y z


  
  
   
7. For what value of λ the equations will have a solutions? Will
the equations have any solution for this value of λ?
2
1;
2 4 ;
4 10
x y z
x y z
x y z


   
  
  
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