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# Appliacation of Matrix.pptx

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# Appliacation of Matrix.pptx

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### Appliacation of Matrix.pptx

1. 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. 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. 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 1/30/2023 Business Mathematics 3
4. 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  1/30/2023 Business Mathematics 4
5. 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: 1/30/2023 Business Mathematics 5
6. 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  1/30/2023 Business Mathematics 6
7. 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     1/30/2023 Business Mathematics 7
8. 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. 1/30/2023 Business Mathematics 8
9. 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  1/30/2023 Business Mathematics 9
10. 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     1/30/2023 Business Mathematics 10
11. 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          1/30/2023 Business Mathematics 11
12. 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          1/30/2023 Business Mathematics 12
13. 13. Practice Question Using matrices, solve the following system of equations x + y + z = 6 x + 2y + 3z = 14 x + 4y + 7z = 30 1/30/2023 Business Mathematics 13
14. 14. Cramer’s Rule • System of Linear Equations • How to solve using Cramer’s Rule 1/30/2023 Business Mathematics 14
15. 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. 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 1/30/2023 Business Mathematics 16 Coeff Matrix
17. 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                    1/30/2023 Business Mathematics 17
18. 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       1/30/2023 Business Mathematics 18
19. 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. 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       1/30/2023 Business Mathematics 20
21. 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                1/30/2023 Business Mathematics 21
22. 22. Row Echelon Form •To be in this form, a matrix must have the following properties. 1/30/2023 Business Mathematics 22
23. 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. 1/30/2023 Business Mathematics 23
24. 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. 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      1/30/2023 Business Mathematics 25
26. 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     1/30/2023 Business Mathematics 26