The document discusses matrices and their applications in solving systems of linear equations. It begins by covering topics like singular and non-singular matrices, and using matrix algebra to write a system of linear equations in matrix form. It then explains how to solve such systems using the inverse of the coefficient matrix, if it is non-singular. Cramer's rule is also introduced as another method to solve systems of up to 3 equations. Examples are provided to illustrate solving systems of equations using matrices. The document concludes by defining row echelon form and providing examples of matrices in both row echelon and reduced row echelon form.

1. School of Management
By: Mr. Manoj Kumar Mishra
1/30/2023 Business Mathematics 1
UNIVERSITY OF STEEL TECHNOLOGY
AND MANAGEMENT
Matrices Algebra
For BBA & B.Com

2. Topic To be Covered
1/30/2023 Business Mathematics 2
1. Solution of Simultaneous Linear Equations
2. Cramer Rule
3. Row echelon form of matrix
4. Gauss Jordan Method

3. Singular Matrix
A square matrix A is said to be singular if lAl = 0 .
A is non-singular if 
A 0.
For Example:
1 1 3
1 3 3
5 3 3

 

 
 
 
Let A=
A is a singular matrix .
A=1(9+9)+1(3+15)+3(3-15)
= 18+18-36
= 0
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4. Non-Singular Matrix
Let B=
1 1 1
2 1 1
1 2 3
 

 
 

 
B is a non-singular matrix.
B= 1(-3+2)-1(6-1)+1(-4+1)
= -1 – 5 – 3
= -9 0

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5. Example -1
Find the value of x for which the matrix is singular.
x 1 0
A = 2 -1 1
3 4 -2
 
 
 
 
 
For matrix A to be singular
   
A = 0
x 1 0
2 -1 1 = 0
3 4 -2
-1 -4 - 3 = 0
7
-2x +7 = 0 x =
2
x 2 4

 
 
Solution:
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6. Solution of Simultaneous Linear Equations
(Matrix Method)
Let the system of 3 linear equations be
1 1 1 1
2 2 2 2
3 3 3 3
a x +b y +c z = d
a x +b y +c z = d
a x +b y + c z = d
This system of linear equation can be written in matrix form as
1
1 1 1
2 2 2 2
3 3 3 3
d
a b c x
a b c y = d
a b c z d
 
   
 
   
 
   
 
   
 
   
 
AX = B ... i

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7. Solution of Simultaneous Linear Equations
(Matrix Method)
1
X = (adjA)B
A

The matrix A is called the coefficient matrix of the
system of linear equations.
Multiplying (i) by A–1, we get
-1
If A 0 i.e. A is non - singular, then A exists.

 
1 1
A AX A B
 

 
1 1
A A X A B
 
 
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8. Important Results
(i) If A is a non-singular matrix, then the system of equations given by
AX = B has a unique solution given by X = A–1B
(ii) If A is a singular matrix and (adjA)B = 0, then the system of
equations given by AX = B is consistent with infinitely many
solutions.
(iii) If A is a singular matrix and (adjA)B  0, then the system of
equations given by AX = B is inconsistent.
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9. Question-1
Using matrix method, solve the following system of linear equations
x + 2y -3z = -4
2x + 3y + 2z = 2
3x - 3y - 4z = 11
Solution:
The given system of equations is
x + 2y - 3z = -4 ...(i)
2x + 3y + 2z = 2 …(ii)
3x -3y - 4z = 11 …(iii)
1 2 -3 x -4
or 2 3 2 y = 2
z
3 -3 -4 11
     
     
     
     
AX = B

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10. Solution (Cont.)
1 2 -3 x -4
where A= 2 3 2 , X= y , B= 2
z
3 -3 -4 11
     
     
     
     
 
2 2 1 3 3 1
-1
1 2 -3
A = 2 3 2
3 -3 -4
1 0 0
= 2 -1 8 Applying C C -2C and C C +3C
3 -9 5
=1(-5+72)=67 0
A exists.
 


ij ij ij
Let C be the cofactor a in A = a , then
 
 
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11. Solution Cont.
 
11 12 13
c =(-12+6) c =- -8-6 c =(-6-9)
=-6 =14 =-15
21 22 23
c =-(-8-9) c =(-4+9) c =-(-3-6)
=17 =5 = 9
31 32 33
c =(4+9) c =-(2+6) c =(3-4)
=13 = -8 = -1
31 32 33
c =(4+9) c =-(2+6) c =(3-4)
=13 = -8 = -1
-6 17 13
14 5 -8
-15 9 -1
 
 
 
 
T
-6 14 -15
adjA = 17 5 9 =
13 -8 -1
 
  
 
 
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12. Solution (Con.)
-1
Now, X= A B
-6 17 13 -4
1
X= 14 5 -8 2
67 -15 9 -1 11
   
    
   
   
x 201 3
1
y = -134 = -2
67
z 67 1
x=3 , y=-2 , z=1
     
      
     
     

-1
-6 17 13
1 1
A = .adj A = 14 5 -8
A 67 -15 9 -1
 
  
 
 
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13. Practice Question
Using matrices, solve the following system of
equations
x + y + z = 6
x + 2y + 3z = 14
x + 4y + 7z = 30
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14. Cramer’s Rule
• System of Linear Equations
• How to solve using Cramer’s Rule
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15. Introduction
• Cramer’s Rule is a method for solving linear
simultaneous equations. It makes use of
determinants and so a knowledge of these is
necessary before proceeding.
• Cramer’s Rule relies on determinants
1/30/2023 Business Mathematics 15

16. Coefficient Matrices
• You can use determinants to solve a system of
linear equations.
• You use the coefficient matrix of the linear
system.
• Linear System
ax+by=e cx+dy=f






d
c
b
a
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Coeff Matrix

17. Using Cramer’s Rule
to Solve a System of Three Equations
Define  
   
11 12 13
21 22 23
31 32 33
1 1
2 2
3 3
a a a
A a a a
a a a
x b
x x and B b
x b
 
 
  
 
 
   
   
   
   
   
   
3
1 2
1 2 3
If 0, then the system has a unique solution
as shown below (Cramer's Rule).
, ,
D
D
D D
x x x
D D D
         
   
    
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18. Using Cramer’s Rule
to Solve a System of Three Equations
where
11 12 13 1 12 13
12 22 23 1 2 22 23
13 32 33 3 32 33
11 1 13 11 12 1
2 12 2 23 3 12 22 2
13 3 33 13 32 3
a a a b a a
D a a a D b a a
a a a b a a
a b a a a b
D a b a D a a b
a b a a a b
  
  
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19. Example 1
Consider the following equations:
    
 
1 2 3
1 2 3
1 2 3
2 4 5 36
3 5 7 7
5 3 8 31
where
2 4 5
3 5 7
5 3 8
x x x
x x x
x x x
A x B
A
  
   
   


 
 
 
 
 

 
1/30/2023 Business Mathematics 19

20. Example 1
   
1
2
3
36
7
31
x
x x and B
x
   
   
   
   
   

   
2 4 5
3 5 7 336
5 3 8
D

   

1
36 4 5
7 5 7 672
31 3 8
D

  
 
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21. Example 1
2
2 36 5
3 7 7 1008
5 31 8
D   
 
3
2 4 36
3 5 7 1344
5 3 31
D

   

1
1
2
2
3
3
672
2
336
1008
3
336
1344
4
336
D
x
D
D
x
D
D
x
D

  

   


  

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22. Row Echelon Form
•To be in this form, a matrix must have the
following properties.
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23. Examples – Row-Echelon Form
•Determine whether each matrix is in row-echelon
form. If it is, determine whether the matrix is in
reduced row-echelon form.
a. b.
c. d.
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24. Examples – Row-Echelon Form
e. f.
•Solution:
•The matrices in (a), (c), (d), and (f) are in row-echelon form.
•The matrices in (d) and (f) are in reduced row-echelon form
because every column that has a leading 1 has zeros in every
position above and below its leading 1.
cont’d
1/30/2023 Business Mathematics 24

25. Using Matrices to Solve Systems of Equations
The use of Elementary Row Operations is required when solving a system of
equations using matrices.
12
7
3
1
13
5
0
2
8
2
6
3



Elementary Row Operations
I. Interchange two rows.
II. Multiply one row by a nonzero number.
III. Add a multiple of one row to a different row.
8
2
6
3
13
5
0
2
12
7
3
1



16
4
12
6
13
5
0
2
12
7
3
1



16
4
12
6
13
5
0
2
12
7
3
1



16
4
12
6
11
19
6
0
12
7
3
1





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26. Use matrices to solve the following systems of equations.
12
2
1
5
15
4
3
2
4
1
1
1










12
2
1
5
7
2
1
0
4
1
1
1





32
7
6
0
7
2
1
0
4
1
1
1





32
7
6
0
7
2
1
0
4
1
1
1




10
5
0
0
7
2
1
0
4
1
1
1




2
1
0
0
7
2
1
0
4
1
1
1




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