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# Matrices

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### Matrices

1. 1. Matrices
2. 2. A matrix is a rectangular array of real numbers. Matrix A has 2 horizontal rows and 3 vertical columns. 3 1 −2  A=  7 −1 0.5  Each entry can be identified by its position in the matrix. 7 is in Row 2 Column 1. -2 is in Row 1 Column 3. A matrix with m rows and n columns is of order m × n. A is of order 2 × 3. If m = n the matrix is said to be a square matrix of order n.Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2
3. 3. Examples: Find the order of each matrix 2 3 1 0  A has three rows and A = 4  2 1 4  four columns.  1  1 6 2  The order of A is 3 × 4.  B has one row and five columns. B = [ 2 5 2 −1 0] The order of B is 1 × 5. B is called a row matrix. 3 1  C=  C is a 2 × 2 square matrix. 6 2Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 3
4. 4. An m × n matrix can be written  a11 a12 L a1n  a a22 L a2 n  A = ai j  =  21    M . M M    am1 am1 am1 amn  Two matrices A = [aij] and B = [bij] are equal if they have the same order and aij = bij for every i and j. 0.5 9  1  For example,  = 2 3  since both matrices 1 7  0.25 7   4      are of order 2 × 2 and all corresponding entries are equal.Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 4
6. 6. Matrices • Diagonal Matrix – A square matrix whose every element other than the diagonal elements are ZERO is called a diagonal matrix. • Diagonal elements – The elements aij are called diagonal elements when i=jCopyright © by Houghton Mifflin Company, Inc. All rights reserved. 6
7. 7. Matrices • Scalar matrix -A diagonal matrix whose diagonal elements are equal • Identity Matrix (Unit matrix) – A diagonal matrix whose diagonal elements are equal to oneCopyright © by Houghton Mifflin Company, Inc. All rights reserved. 7
8. 8. Matrices • Triangular matrix – A square matrix whose elements below or above the diagonal are zero. • What is an upper triangular matrix? • What is a lower triangular matrix?Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 8
9. 9. To add matrices: 1. Check to see if the matrices have the same order. 2. Add corresponding entries. Example: Find the sums A + B and B + C. 1 5  A= 2 1  B =  2 0 6  C = 3 −3 0    −1 0 −3 3 2 4  0 6        A has order 3 × 2 and B has order 2 × 3. So they cannot be added. C has order 2 × 3 and can be added to B.  2 0 6  3 −3 0  5 −3 6  B+C =   + 3 2 4  =  2 2 1   −1 0 −3    Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 9
10. 10. To subtract matrices: 1. Check to see if the matrices have the same order. 2. Subtract corresponding entries. Example: Find the differences A – B and B – C. 3 7   2 −1  −1 5 1  A=  B =  4 −5 C =  2 1 6  2 1      A and B are both of order 2 × 2 and can be subtracted.  3 7   2 −1  1 8  A−B =   −  4 −5 =  −2 6  2 1      Since B is of order 2 × 2 and C is of order 3 × 2, they cannot be subtracted.Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 10
11. 11. If A = [aij] is an m × n matrix and c is a scalar (a real number), then the m × n matrix cA = [caij] is the scalar multiple of A by c.  2 5 −1 3 4 0  Example: Find 2A and –3A for A =  . 2 7  2   2(2) 2(5) 2( −1)   4 10 −2  2 A =  2(3) 2(4) 2(0)  = 6 8 0       2(2) 2(7) 2(2)   4 14 4       −3(2) −3(5) −3( −1)   −6 −15 3  1 − A =  −3(3) −3(4) −3(0)  =  −9 −12 0      3  −3(2) −3(7) −3(2)   −6 −21 −6     Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 11
12. 12. Example: Calculate the value of 3A – 2B + C with  2 −1 5 2 5 2 A = 3 5  B = 1 0  and C = 1 0         4 −2     3 −1     3 −1     2 −1 5 2  5 2 3 A − 2 B + C = 3  3 5  − 2 1 0  +  1 0         4 −2     3 − 1   3 −1       6 −3  10 4  5 2  1 − 5 = 9 15 −  2    0  + 1 0  =  8 15       12 −6   6 −2   3 −1   9 − 5         Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 12
13. 13. Multiplication of Matrices • The product AB of two matrices A and B is defined only when the number of columns A is same as the number of rows of B. • A = m x n matrix • B = n x p matrix. • The order of AB is m x p.Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 13
14. 14. Multiplication of Matrices • AB may not be equal to BA • If product AB is defined product BA may not be defined. • If A is a square matrix then A can be multiplied by itself.Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 14
15. 15. Transpose of a Matrix • Let A be a matrix. The matrix obtained by interchanging the rows and columns is called the transpose of the matrix A.Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 15
16. 16. Symmetric and Skew Symmetric Matrices • For a square matrix A if A=AT then it is a symmetric matrix. For a square matrix A if A= -AT then it is a skew symmetric matrix. Square matrix A + AT symmetric Square matrix A – AT is skew symmetricCopyright © by Houghton Mifflin Company, Inc. All rights reserved. 16
17. 17. • If square matrices AB= BA then A,B are commutative • If square matrices AB=-BA the A,B are anti commutative • If A2 = A then A is idempotentCopyright © by Houghton Mifflin Company, Inc. All rights reserved. 17
18. 18. Determinant of a matrix • The square matrix A has a uniquely determined determinant associated with the matrix. • The determinant of a product of a matrix is the product of their determinants.Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 18
19. 19. Singular and Non singular matrices • A square matrix A is singular if determinant A is zero. • A square matrix A is non singular is determinant A is not equal to Zero.Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 19
20. 20. Elementary Operations in matrices. 1. Interchange two rows or columns of a matrix. 2. Multiply a row or column of a matrix by a non zero constant. 3. Add a multiple of one row or column of a matrix to another.Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 20
21. 21. What is the use of Elementary operations • A sequence of elementary row operations transforms the matrix of a system into the matrix of another system with the same solutions as the original system. • Take matrix A X B = AB • If we make elementary row operation in AB then it is equivalent to making the same operation in A and multiplying it with B.Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 21
22. 22. An augmented matrix and a coefficient matrix are associated with each system of linear equations. 2 x + 3 y − z = 12 For the system   x − 8y = 16 2 3 - 1 12 The augmented matrix is  . 1 - 8 0 16  2 3 −1 The coefficient matrix is  . 1 −8 0 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 22
23. 23. Example: Apply the elementary row operation R1 ↔ R2 to the augmented matrix of the system  x + 2 y = 8 .  3x − y = 10 Row Operation Augmented Matrix System 1 2 8  x + 2y = 8 3   -1 10   3x − y = 10 R1 ↔ R2 ↓ ↓ 3 -1 10 3x − y = 10 1    2 8  x + 2y = 8 Note that the two systems are equivalent.Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 23
24. 24. Example: Apply the elementary row operation 3R2 to the augmented matrix of the system  x + 2 y = 8 .  3x − y = 10 Row Operation Augmented Matrix System 1 2 8  x + 2y = 8 3   -1 10   3x − y = 10 3R2 ↓ ↓ 1 2 8  9 -3 30  Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 24
25. 25. Example: Apply the row operation –3R1 + R2 to the augmented matrix of the system  x + 2 y = 8 .  3x − y = 10 Row Operation Augmented Matrix System 1 2 8  x + 2y = 8 3   -1 10   3x − y = 10 –3R1 + R2 ↓ ↓ 1 2 8  x + 2y = 8 0    -7 - 14 −7 y = −14Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 25