matrices

1. B.L.D.E.Association’s
SHREE SANGANABASAVA MAHASWAMIJI POLYTECHNIC, VIJAYAPUR - 03
SCIENCE DEPARTMENT
Course: Engineering Mathematics 20SC01T
LE/SC

2. B.L.D.E.Association’s
SHREE SANGANABASAVA MAHASWAMIJI POLYTECHNIC VIJAYAPUR – 03
MECHANICAL ENGINEERING DEPARTMENT
B.L.D.E.Association’s
SHREE SANGANABASAVA MAHASWAMIJI POLYTECHNIC VIJAYAPUR – 03
SCIENCE DEPARTMENT
Course : Engineering Course Coordinator : Shri .
INVERSE AND APPLICATIONS OF MATRICES
Singular and non singular matrices:
 Singular matrix: A square matrix is said to be singular if its
determinant is equal to zero. i.e

A 0

3 2 9 3
1. Let A 2. Let A
6 4 15 5
3 2 9 3
Then A Then A
6 4 15 5
12 12 0 45 45 0
A is Singular matrix
   
 
   
   
 
     


3. B.L.D.E.Association’s
SHREE SANGANABASAVA MAHASWAMIJI POLYTECHNIC VIJAYAPUR – 03
MECHANICAL ENGINEERING DEPARTMENT
B.L.D.E.Association’s
SHREE SANGANABASAVA MAHASWAMIJI POLYTECHNIC VIJAYAPUR – 03
SCIENCE DEPARTMENT
Course : Engineering Course Coordinator : Shri .
     
     
1 2 3
A= 1 0 4
2 4 6
1 2 3
1 0 4
2 4 6
1 0 16 2 6 8 3 4 0
1 16 2 2 3 4
16 4 12 0
Ais singular matrix
Evaluate
let
 
 
 
 
 
     
    
    


4. B.L.D.E.Association’s
SHREE SANGANABASAVA MAHASWAMIJI POLYTECHNIC VIJAYAPUR – 03
MECHANICAL ENGINEERING DEPARTMENT
B.L.D.E.Association’s
SHREE SANGANABASAVA MAHASWAMIJI POLYTECHNIC VIJAYAPUR – 03
SCIENCE DEPARTMENT
Course : Engineering Course Coordinator : Shri .
Non-Singular matrix
 Singular matrix : A square matrix is said to be non-singular if its determinant
is not equal to zero. i.e A 0

2 1 4 2
1. Let A 2. Let A
5 6 5 3
2 1 4 2
Then A Then A
5 6 5 3
12 5 7 12 10 2
A is non-singularmatrix
   
 
   
   
 
     


5. B.L.D.E.Association’s
SHREE SANGANABASAVA MAHASWAMIJI POLYTECHNIC VIJAYAPUR – 03
MECHANICAL ENGINEERING DEPARTMENT
B.L.D.E.Association’s
SHREE SANGANABASAVA MAHASWAMIJI POLYTECHNIC VIJAYAPUR – 03
SCIENCE DEPARTMENT
Course : Engineering Course Coordinator : Shri .
MINORS AND COFACTORS OF AN ELEMENT
Minor of an element: Minor of an element is the determinant obtained by deleting
row and column containing that element from the given matrix.
11 12 13
21 22 23
31 32 33
22 23
11
32 33
21 23
12
31 33
12 13
31
22 23
Example: A
minor of a
minor of a
minor of a
a a a
a a a
a a a
a a
a a
a a
a a
a a
a a
 
 
  
 
 




6. B.L.D.E.Association’s
SHREE SANGANABASAVA MAHASWAMIJI POLYTECHNIC VIJAYAPUR – 03
MECHANICAL ENGINEERING DEPARTMENT
B.L.D.E.Association’s
SHREE SANGANABASAVA MAHASWAMIJI POLYTECHNIC VIJAYAPUR – 03
SCIENCE DEPARTMENT
Course : Engineering Course Coordinator : Shri .
COFACTORS OF AN ELEMENT:
 
j ij
i+j
ij
i
ij
c
andisdefined asc = -1 minor of
sign of cofatorsof an element can beremembered asfollo
ctor of an element a in thesquare matrix A is denoted by
a
the ws
Cofa
  
  
  

7. B.L.D.E.Association’s
SHREE SANGANABASAVA MAHASWAMIJI POLYTECHNIC VIJAYAPUR – 03
MECHANICAL ENGINEERING DEPARTMENT
B.L.D.E.Association’s
SHREE SANGANABASAVA MAHASWAMIJI POLYTECHNIC VIJAYAPUR – 03
SCIENCE DEPARTMENT
Course : Engineering Course Coordinator : Shri .
 
 
 
11 12 13
21 22 23
31 32 33
1 1 22 23 22 23
11
32 33 32 33
1 2 21 23 21 23
12
31 33 31 33
3 1 12 13 12 13
31
22 23 22 23
Example: A
cofactor of a 1
cofactorof a 1
cofactor of a 1
a a a
a a a
a a a
a a a a
a a a a
a a a a
a a a a
a a a a
a a a a



 
 
  
 
 
   
   
   

8. B.L.D.E.Association’s
SHREE SANGANABASAVA MAHASWAMIJI POLYTECHNIC VIJAYAPUR – 03
MECHANICAL ENGINEERING DEPARTMENT
B.L.D.E.Association’s
SHREE SANGANABASAVA MAHASWAMIJI POLYTECHNIC VIJAYAPUR – 03
SCIENCE DEPARTMENT
Course : Engineering Course Coordinator : Shri .
Adjoint of asquerematrix:Adjoint of asquerematrixisdefinedas thetranspose
of thematrix formed by thecorrespondingcofators.andisdenotedisbyadjA.
   
   
 
1 1 5 2
Example:1).A , 2). A ,
6 2 3 1
1 1 5 2
:A :A
6 2 3 1
1 2 2 5 1 1
1 6 6 2 3 3
6 1 1
find adjA find adjA
Solution Solution
cofactor of cofactor of
cofactor of cofactor of
cofactor of cofacto

 
  

 
  
       
        
    
   
   
   
   
   
   
 
   
3 2 2
2 1 1 1 5 5
2 6 1 3
1 1 2 5
2 1 1 2
Adj.A Adj.A
6 1 3 5
r of
cofactor of cofactor of
cofactor matrix cofactor matrix
    
      

 

 
 
   
   
   
   
   
   

9. B.L.D.E.Association’s
SHREE SANGANABASAVA MAHASWAMIJI POLYTECHNIC VIJAYAPUR – 03
MECHANICAL ENGINEERING DEPARTMENT
B.L.D.E.Association’s
SHREE SANGANABASAVA MAHASWAMIJI POLYTECHNIC VIJAYAPUR – 03
SCIENCE DEPARTMENT
Course : Engineering Course Coordinator : Shri .
   
   
   
2 1 2 3
Example:3).A , 4). A ,
3 4 1 4
2 1 2 3
:A :A
3 4 1 4
2 4 4 2 4 4
1 3 3 3 1 1
3 1 1 1 3
find adjA find adjA
Solution Solution
cofactor of cofactor of
cofactor of cofactor of
cofactor of cofactor of

 

 
     
        
       
   
   
   
   
   
   
   
3
4 2 2 4 2 2
4 1
4 3
3 2
1 2
4 3
4 1
Adj.A Adj.A
1 2
3 2
cofactor of cofactor of
cofactor matrix cofactor matrix
 
 
 
 
 
 
 
 
     


 


 


 
 
 
 
 
 

10. B.L.D.E.Association’s
SHREE SANGANABASAVA MAHASWAMIJI POLYTECHNIC VIJAYAPUR – 03
MECHANICAL ENGINEERING DEPARTMENT
B.L.D.E.Association’s
SHREE SANGANABASAVA MAHASWAMIJI POLYTECHNIC VIJAYAPUR – 03
SCIENCE DEPARTMENT
Course : Engineering Course Coordinator : Shri .
Example:
1 4 2 3
5. 6.
2 1 4 1
1 2 2 3
7. 8.
2 1 2 5
3 2 1 2
9. 10.
2 1 3 4
find adjA
   
   
 
   
   
   

   
 
   
   

   

11. B.L.D.E.Association’s
SHREE SANGANABASAVA MAHASWAMIJI POLYTECHNIC VIJAYAPUR – 03
MECHANICAL ENGINEERING DEPARTMENT
B.L.D.E.Association’s
SHREE SANGANABASAVA MAHASWAMIJI POLYTECHNIC VIJAYAPUR – 03
SCIENCE DEPARTMENT
Course : Engineering Course Coordinator : Shri .
   
 
1 2 3
Example: 1). 2 4 5
3 5 6
1 2 3
: 2 4 5
3 5 6
4 5 2 3
cofactor of 1= + = + 24-25 = -1 cofactor of 3= + = + 10-12 = -2
5 6 4 5
2 5 1 3
cofactor of 2= - = - 12-15 = -3 cofactor of 5= -
3 6 2
find adjA of
solution A
 
 
 
 
 
   
   
   
   
   
  
   
  
 
   
 
 
= - 5-6 = 1
5
2 4 1 2
cofactor of 3= + = + 10-12 = -2 cofactor of 6= + = + 4 4 = 0
3 5 2 4
1 3 2
2 3
cofactor of 2= - = - 12-15 = 3 cofactor of matrix A= 3 3 1
5 6
2 1 0
1 3 2
1 3
cofactor of 4= + = + 6-9 = -3 Adj.A= 3 3 1
3 6
2 1 0
cofact
 
 
 
 
 
 
 
 
 
 

 


 


 
1 2
or of 5= - = - 5-6 = 1
3 5

12. B.L.D.E.Association’s
SHREE SANGANABASAVA MAHASWAMIJI POLYTECHNIC VIJAYAPUR – 03
MECHANICAL ENGINEERING DEPARTMENT
B.L.D.E.Association’s
SHREE SANGANABASAVA MAHASWAMIJI POLYTECHNIC VIJAYAPUR – 03
SCIENCE DEPARTMENT
Course : Engineering Course Coordinator : Shri .
   
 
1 2 3
Example: 1). 1 3 3
2 4 3
1 2 3
: 1 3 3
2 4 3
3 3 2 3
cofactor of 1 = + = + 9 12 =-3 cofactor of 2 = + = + 6 9 = -3
4 3 3 3
1 3 1 3
cofactor of 2 = - = - 3 6 = 3 cofactor of 4 = - = - 3
2 3 1 3
find adjA of
solution A
 
 
 
 
 
   
   
   
   
   
  
   
  
 
  
   
 
 
3 =0
1 3 1 2
cofactor of 3 = + = + 4 6 = -2 cofactor of 3= + = + 3 2 = 1
2 4 1 3
3 3 2
2 3
cofactor of 1 = - = - 6 12 = 6 cofactor of matrix A= 6 3 0
4 3
3 0 1
3 6 3
1 3
cofactor of 3 = + = + 3 6 = -3 Adj.A= 3 3 0
2 3
2 0 1
cofactor of 3 =
 
 
 
 
 
 
 
 
 
 

 
 
 

 
 

 
1 2
- = - 4 4 = 0
2 4


13. B.L.D.E.Association’s
SHREE SANGANABASAVA MAHASWAMIJI POLYTECHNIC VIJAYAPUR – 03
MECHANICAL ENGINEERING DEPARTMENT
B.L.D.E.Association’s
SHREE SANGANABASAVA MAHASWAMIJI POLYTECHNIC VIJAYAPUR – 03
SCIENCE DEPARTMENT
Course : Engineering Course Coordinator : Shri .
Example:
5 3 1 1 2 2
3. 2 1 0 4. 1 3 0
4 2 1 0 2 1
1 1 2 1 2 0
5. 2 1 1 6. 1 2 3
4 1 2 3 0 4
1 2 1 1 0 2
7. 1 1 2 8. 2 1 0
2 1 1 3 2 1
 
   
   

   
   
 
   
 
   
   

   
   
  
   

   
   

   
   

   

14. B.L.D.E.Association’s
SHREE SANGANABASAVA MAHASWAMIJI POLYTECHNIC VIJAYAPUR – 03
MECHANICAL ENGINEERING DEPARTMENT
B.L.D.E.Association’s
SHREE SANGANABASAVA MAHASWAMIJI POLYTECHNIC VIJAYAPUR – 03
SCIENCE DEPARTMENT
Course : Engineering Course Coordinator : Shri .
Inverse of a matrix :
Let A be a non - singular square matrix.If there exists a square matrix B, such that AB = BA = I,
where I is the unit matrix of the same order as that of A, then Bis called the inve
 
rse of A and it is denoted
-1
by A .This can be determined by using the formula.
1
-1
A = adjA
A

15. B.L.D.E.Association’s
SHREE SANGANABASAVA MAHASWAMIJI POLYTECHNIC VIJAYAPUR – 03
MECHANICAL ENGINEERING DEPARTMENT
B.L.D.E.Association’s
SHREE SANGANABASAVA MAHASWAMIJI POLYTECHNIC VIJAYAPUR – 03
SCIENCE DEPARTMENT
Course : Engineering Course Coordinator : Shri .
2 1 3 1
A A
1 2 2 4
A A
2 1 4 1
1 2 2 3
1
-1
= .adj
A
Example:1 ).Find the inverse of the matrix 2) Find the inverse of the matrix
2 1
,
: : =
=4 - 1 = 3 = 12 - 2 = 10
adjA = adjA =
A
3 1
1 2 2 4
Solution Solution


   

   

   


 
 
 
 

   
1
-1
.A = .adj.A
A
2 1 4 1
1 1
-1 -1
= . = .
3 10
1 2 2 3
A
A A
   

   

   

16. B.L.D.E.Association’s
SHREE SANGANABASAVA MAHASWAMIJI POLYTECHNIC VIJAYAPUR – 03
MECHANICAL ENGINEERING DEPARTMENT
B.L.D.E.Association’s
SHREE SANGANABASAVA MAHASWAMIJI POLYTECHNIC VIJAYAPUR – 03
SCIENCE DEPARTMENT
Course : Engineering Course Coordinator : Shri .
     
 
1 2 3
1).find the inverse of A = 3 1 2
2 3 1
1 2 3 1 2 3
solution: A = 3 1 2 , A = 3 1 2
2 3 1 2 3 1
=1 1-6 -2 3-4 +3 9-2
= -5+2+21=18
1 2
cofactor of 1 = + = + 1-6 = -5 cofactor of 2 = +
3 1
 
 
 
 
 
 
 
 
 
 
 
   
   
 
2 3
= + 4-3 = 1
1 2
3 2 1 3
cofactor of 2 = - = - 3-4 = 1 cofactor of 3 = - = - 2-9 = 7
2 1 3 2
3 1 1 2
cofactor of 3 = + = + 9-2 = 7 cofactor of 1= + = + 1-6 = -5
2 3 3 1
-5 7 1
2 3
cofactor of 3 = - = - 2-9 = 7 adjA= 1 -5 7
3 1
7 1 -5
1 3
cofactor of 1 = + =
2 1
 
 
 
 
 
 
 
1
-1
+ 1-6 = -5 . .
-5 7 1
1 2 1
-1
2 - - 3 - 4 1 . 1 -5 7
18
2 3
7 1 -5
A adj A
A
cofactor of A
 
 
 
 
 

   

17. B.L.D.E.Association’s
SHREE SANGANABASAVA MAHASWAMIJI POLYTECHNIC VIJAYAPUR – 03
MECHANICAL ENGINEERING DEPARTMENT
B.L.D.E.Association’s
SHREE SANGANABASAVA MAHASWAMIJI POLYTECHNIC VIJAYAPUR – 03
SCIENCE DEPARTMENT
Course : Engineering Course Coordinator : Shri .
     
   
 
1 1 1
2).find the inverse of A = 2 1 3
3 2 4
1 1 1
solution: A = 2 1 3
3 2 4
=1 4 6 1 8 9 +1 4 3
= 2+17-7=12
cofactor of 1 = + 4 6 = 2 cofactor of 3 = + 3 1 = - 2
cofactor of 1 = - 8 9 = -17


 


 
     
    
 
 
 
 
 
 
 
   
 
 
 
cofactor of 2 = - 3 2 = - 1
cofactor of 1 = + 4 3 = -7 cofactor of 4 = + 1 2 = 1
2 2 2
cofactor of 2 = - 4 2 = 2 adjA= 17 7 1
7 5 1
1
-1
cofactor of 1= + 4 3 = 7 . .
2 2
1
-1
3 2 3 5 .
12
A adj A
A
cofactor of A
 
   

   

  

     
 
 
 
 
 
2
17 7 1
7 5 1
 

 
 
 
 
 

18. B.L.D.E.Association’s
SHREE SANGANABASAVA MAHASWAMIJI POLYTECHNIC VIJAYAPUR – 03
MECHANICAL ENGINEERING DEPARTMENT
B.L.D.E.Association’s
SHREE SANGANABASAVA MAHASWAMIJI POLYTECHNIC VIJAYAPUR – 03
SCIENCE DEPARTMENT
Course : Engineering Course Coordinator : Shri .
     
   
 
1 2 3
3).find the inverse of A = 1 3 3
2 4 3
1 2 3
solution: A = 1 3 3
2 4 3
=1 9 12 2 3 6 +3 4 6
= - 3 + 6 - 6 = -3
cofactor of 1 = + 9 12 = -3 cofactor of 2 = + 6 9 = - 3
cofactor of 2 = - 3 6 = 3 cofa
   
 

 
 
 
 
 
 
   
 
 
 
ctor of 4 = - 3 3 = 0
cofactor of 3 = + 4 6 = -2 cofactor of 3 = + 3 2 = 1
3 6 3
cofactor of 1 = - 6 12 = 6 adjA= 3 3 0
2 0 1
1
-1
cofactor of 3= + 3 6 = -3 . .
3 6 3
1
-1
3 4 4 0 . 3 3 0
3
2 0 1
A adj A
A
cofactor of A

 
 
 

 
 
     


 
 
 
 
 





 
 


19. B.L.D.E.Association’s
SHREE SANGANABASAVA MAHASWAMIJI POLYTECHNIC VIJAYAPUR – 03
MECHANICAL ENGINEERING DEPARTMENT
Course : Engineering
B.L.D.E.Association’s
SHREE SANGANABASAVA MAHASWAMIJI POLYTECHNIC VIJAYAPUR – 03
SCIENCE DEPARTMENT
Thank You
For Any Questions / Suggestions please Contact
Shri . Prakash.k.Ummaraji Lecturer
Science Department
B.L.D.E.A’s
Shree Sanganabasava Mahaswamiji Polytechnic,
Ashram road, Vijayapur – 03
Mobile : 9480116234
https://www.facebook.com/sudhanva.kulkarni.14 9480116234
Course Coordinator : Shri .