Matrices ,Basics, Determinant, Inverse, EigenValues, Linear Equations, RANK
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The document provides an overview of matrix algebra, covering key concepts such as definitions, types of matrices, operations including addition, subtraction, scalar multiplication, and multiplication of matrices. It also discusses properties of matrices, determinants, and concepts like symmetry and transposition. This foundational information is essential for understanding systems of linear equations and mathematical treatments involving matrices.
![Matrices - Operations
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Matrices ,Basics, Determinant, Inverse, EigenValues, Linear Equations, RANK
- 2. Matrices - Introduction Matrix algebra has at least two advantages: •Reduces complicated systems of equations to simple expressions •Adaptable to systematic method of mathematical treatment and well suited to computers Definition: A matrix is a set or group of numbers arranged in a square or rectangular array enclosed by two brackets 1 1 0 3 2 4 d c b a
- 3. Matrices - Introduction Properties: •A specified number of rows and a specified number of columns •Two numbers (rows x columns) describe the dimensions or size of the matrix. Examples: 3x3 matrix 2x4 matrix 1x2 matrix 3 3 3 5 1 4 4 2 1 2 3 3 3 0 1 0 1 1 1
- 4. Matrices - Introduction TYPES OF MATRICES 1. Column matrix or vector: The number of rows may be any integer but the number of columns is always 1 2 4 1 3 1 1 21 11 m a a a
- 5. Matrices - Introduction TYPES OF MATRICES 2. Row matrix or vector Any number of columns but only one row 6 1 1 2 5 3 0 n a a a a 1 13 12 11
- 6. Matrices - Introduction TYPES OF MATRICES 3. Rectangular matrix Contains more than one element and number of rows is not equal to the number of columns 6 7 7 7 7 3 1 1 0 3 3 0 2 0 0 1 1 1 n m
- 7. Matrices - Introduction TYPES OF MATRICES 4. Square matrix The number of rows is equal to the number of columns (a square matrix A has an order of m) 0 3 1 1 1 6 6 0 9 9 1 1 1 m x m The principal or main diagonal of a square matrix is composed of all elements aij for which i=j
- 8. Matrices - Introduction TYPES OF MATRICES 5. Diagonal matrix A square matrix where all the elements are zero except those on the main diagonal 1 0 0 0 2 0 0 0 1 9 0 0 0 0 5 0 0 0 0 3 0 0 0 0 3 i.e. aij =0 for all i = j aij = 0 for some or all i = j
- 9. Matrices - Introduction TYPES OF MATRICES 6. Unit or Identity matrix - I A diagonal matrix with ones on the main diagonal 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 1 0 0 1 i.e. aij =0 for all i = j aij = 1 for some or all i = j ij ij a a 0 0
- 10. Matrices - Introduction TYPES OF MATRICES 7. Null (zero) matrix - 0 All elements in the matrix are zero 0 0 0 0 0 0 0 0 0 0 0 0 0 ij a For all i,j
- 11. Matrices - Introduction TYPES OF MATRICES 8. Triangular matrix A square matrix whose elements above or below the main diagonal are all zero 3 2 5 0 1 2 0 0 1 3 2 5 0 1 2 0 0 1 3 0 0 6 1 0 9 8 1
- 12. Matrices - Introduction TYPES OF MATRICES 8a. Upper triangular matrix A square matrix whose elements below the main diagonal are all zero i.e. aij = 0 for all i > j 3 0 0 8 1 0 7 8 1 3 0 0 0 8 7 0 0 4 7 1 0 4 4 7 1 ij ij ij ij ij ij a a a a a a 0 0 0
- 13. Matrices - Introduction TYPES OF MATRICES A square matrix whose elements above the main diagonal are all zero 8b. Lower triangular matrix i.e. aij = 0 for all i < j 3 2 5 0 1 2 0 0 1 ij ij ij ij ij ij a a a a a a 0 0 0
- 14. Matrices – Introduction TYPES OF MATRICES 9. Scalar matrix A diagonal matrix whose main diagonal elements are equal to the same scalar A scalar is defined as a single number or constant 1 0 0 0 1 0 0 0 1 6 0 0 0 0 6 0 0 0 0 6 0 0 0 0 6 i.e. aij = 0 for all i = j aij = a for all i = j ij ij ij a a a 0 0 0 0 0 0
- 15. Matrices - Operations EQUALITY OF MATRICES Two matrices are said to be equal only when all corresponding elements are equal Therefore their size or dimensions are equal as well 3 2 5 0 1 2 0 0 1 3 2 5 0 1 2 0 0 1 A = B = A = B
- 16. Matrices - Operations Some properties of equality: •IIf A = B, then B = A for all A and B •IIf A = B, and B = C, then A = C for all A, B and C 3 2 5 0 1 2 0 0 1 A = B = 33 32 31 23 22 21 13 12 11 b b b b b b b b b If A = B then ij ij b a
- 17. Matrices - Operations ADDITION AND SUBTRACTION OF MATRICES The sum or difference of two matrices, A and B of the same size yields a matrix C of the same size ij ij ij b a c Matrices of different sizes cannot be added or subtracted
- 18. Matrices - Operations SCALAR MULTIPLICATION OF MATRICES Matrices can be multiplied by a scalar (constant or single element) Let k be a scalar quantity; then kA = Ak Ex. If k=4 and 1 4 3 2 1 2 1 3 A
- 19. Matrices - Operations 4 16 12 8 4 8 4 12 4 1 4 3 2 1 2 1 3 1 4 3 2 1 2 1 3 4 Properties: • k (A + B) = kA + kB • (k + g)A = kA + gA • k(AB) = (kA)B = A(k)B • k(gA) = (kg)A
- 20. Matrices - Operations MULTIPLICATION OF MATRICES The product of two matrices is another matrix Two matrices A and B must be conformable for multiplication to be possible i.e. the number of columns of A must equal the number of rows of B Example. A x B = C (1x3) (3x1) (1x1)
- 21. Matrices - Operations B x A = Not possible! (2x1) (4x2) A x B = Not possible! (6x2) (6x3) Example A x B = C (2x3) (3x2) (2x2)
- 22. Matrices - Operations 22 21 12 11 32 31 22 21 12 11 23 22 21 13 12 11 c c c c b b b b b b a a a a a a 22 32 23 22 22 12 21 21 31 23 21 22 11 21 12 32 13 22 12 12 11 11 31 13 21 12 11 11 ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( c b a b a b a c b a b a b a c b a b a b a c b a b a b a Successive multiplication of row i of A with column j of B – row by column multiplication
- 23. Matrices - Operations ) 3 7 ( ) 2 2 ( ) 8 4 ( ) 5 7 ( ) 6 2 ( ) 4 4 ( ) 3 3 ( ) 2 2 ( ) 8 1 ( ) 5 3 ( ) 6 2 ( ) 4 1 ( 3 5 2 6 8 4 7 2 4 3 2 1 57 63 21 31 Remember also: IA = A 1 0 0 1 57 63 21 31 57 63 21 31
- 24. Matrices - Operations Assuming that matrices A, B and C are conformable for the operations indicated, the following are true: 1. AI = IA = A 2. A(BC) = (AB)C = ABC - (associative law) 3. A(B+C) = AB + AC - (first distributive law) 4. (A+B)C = AC + BC - (second distributive law) Caution! 1. AB not generally equal to BA, BA may not be conformable 2. If AB = 0, neither A nor B necessarily = 0 3. If AB = AC, B not necessarily = C
- 25. Matrices - Operations If AB = 0, neither A nor B necessarily = 0 0 0 0 0 3 2 3 2 0 0 1 1
- 26. Matrices - Operations To transpose: Interchange rows and columns The dimensions of AT are the reverse of the dimensions of A 1 3 5 7 4 2 3 2A A 1 7 3 4 5 2 2 3 T T A A 2 x 3 3 x 2
- 27. Matrices - Operations Properties of transposed matrices: 1. (A+B)T = AT + BT 2. (AB)T = BT AT 3. (kA)T = kAT 4. (AT)T = A
- 28. Matrices - Operations SYMMETRIC MATRICES A Square matrix is symmetric if it is equal to its transpose: A = AT d b b a A d b b a A T
- 29. Matrices - Operations DETERMINANT OF A MATRIX To compute the inverse of a matrix, the determinant is required Each square matrix A has a unit scalar value called the determinant of A, denoted by det A or |A| 5 6 2 1 5 6 2 1 A A If then
- 30. Matrices - Operations If A = [A] is a single element (1x1), then the determinant is defined as the value of the element Then |A| =det A = a11 If A is (n x n), its determinant may be defined in terms of order (n-1) or less.
- 31. Matrices - Operations MINORS If A is an n x n matrix and one row and one column are deleted, the resulting matrix is an (n-1) x (n-1) submatrix of A. The determinant of such a submatrix is called a minor of A and is designated by mij , where i and j correspond to the deleted row and column, respectively. mij is the minor of the element aij in A.
- 32. Matrices - Operations 33 32 31 23 22 21 13 12 11 a a a a a a a a a A Each element in A has a minor Delete first row and column from A . The determinant of the remaining 2 x 2 submatrix is the minor of a11 eg. 33 32 23 22 11 a a a a m
- 33. Matrices - Operations Therefore the minor of a12 is: And the minor for a13 is: 33 31 23 21 12 a a a a m 32 31 22 21 13 a a a a m
- 34. Matrices - Operations COFACTORS The cofactor Cij of an element aij is defined as: ij j i ij m C ) 1 ( When the sum of a row number i and column j is even, cij = mij and when i+j is odd, cij =-mij 13 13 3 1 13 12 12 2 1 12 11 11 1 1 11 ) 1 ( ) 3 , 1 ( ) 1 ( ) 2 , 1 ( ) 1 ( ) 1 , 1 ( m m j i c m m j i c m m j i c
- 35. Matrices - Operations DETERMINANTS CONTINUED The determinant of an n x n matrix A can now be defined as n nc a c a c a A A 1 1 12 12 11 11 det The determinant of A is therefore the sum of the products of the elements of the first row of A and their corresponding cofactors. (It is possible to define |A| in terms of any other row or column but for simplicity, the first row only is used)
- 36. Matrices - Operations Therefore the 2 x 2 matrix : 22 21 12 11 a a a a A Has cofactors : 22 22 11 11 a a m c And: 21 21 12 12 a a m c And the determinant of A is: 21 12 22 11 12 12 11 11 a a a a c a c a A
- 37. Matrices - Operations Example 1: 2 1 1 3 A 5 ) 1 )( 1 ( ) 2 )( 3 ( A
- 38. Matrices - Operations For a 3 x 3 matrix: 33 32 31 23 22 21 13 12 11 a a a a a a a a a A The cofactors of the first row are: 31 22 32 21 32 31 22 21 13 31 23 33 21 33 31 23 21 12 32 23 33 22 33 32 23 22 11 ) ( a a a a a a a a c a a a a a a a a c a a a a a a a a c
- 39. Matrices - Operations The determinant of a matrix A is: 21 12 22 11 12 12 11 11 a a a a c a c a A Which by substituting for the cofactors in this case is: ) ( ) ( ) ( 31 22 32 21 13 31 23 33 21 12 32 23 33 22 11 a a a a a a a a a a a a a a a A
- 40. Matrices - Operations Example 2: 1 0 1 3 2 0 1 0 1 A 4 ) 2 0 )( 1 ( ) 3 0 )( 0 ( ) 0 2 )( 1 ( A ) ( ) ( ) ( 31 22 32 21 13 31 23 33 21 12 32 23 33 22 11 a a a a a a a a a a a a a a a A
- 41. Matrices - Operations ADJOINT MATRICES A cofactor matrix C of a matrix A is the square matrix of the same order as A in which each element aij is replaced by its cofactor cij . Example: 4 3 2 1 A 1 2 3 4 C If The cofactor C of A is
- 42. Matrices - Operations The adjoint matrix of A, denoted by adj A, is the transpose of its cofactor matrix T C adjA It can be shown that: A(adj A) = (adjA) A = |A| I Example: 1 3 2 4 10 ) 3 )( 2 ( ) 4 )( 1 ( 4 3 2 1 T C adjA A A
- 43. Matrices - Operations I adjA A 10 10 0 0 10 1 3 2 4 4 3 2 1 ) ( I A adjA 10 10 0 0 10 4 3 2 1 1 3 2 4 ) (
- 44. Matrices - Operations USING THE ADJOINT MATRIX IN MATRIX INVERSION A adjA A 1 Since AA-1 = A-1 A = I and A(adj A) = (adjA) A = |A| I then
- 45. Matrices - Operations Example 1 . 0 3 . 0 2 . 0 4 . 0 1 3 2 4 10 1 1 A 4 3 2 1 A = To check AA-1 = A-1 A = I I A A I AA 1 0 0 1 4 3 2 1 1 . 0 3 . 0 2 . 0 4 . 0 1 0 0 1 1 . 0 3 . 0 2 . 0 4 . 0 4 3 2 1 1 1
- 46. Matrices - Operations Example 2 1 2 1 0 1 2 1 1 3 A |A| = (3)(-1-0)-(-1)(-2-0)+(1)(4-1) = -2 ), 1 ( ), 1 ( ), 1 ( 31 21 11 c c c The determinant of A is The elements of the cofactor matrix are ), 2 ( ), 4 ( ), 2 ( 32 22 12 c c c ), 5 ( ), 7 ( ), 3 ( 33 23 13 c c c
- 47. Matrices - Operations 5 2 1 7 4 1 3 2 1 C The cofactor matrix is therefore so 5 7 3 2 4 2 1 1 1 T C adjA and 5 . 2 5 . 3 5 . 1 0 . 1 0 . 2 0 . 1 5 . 0 5 . 0 5 . 0 5 7 3 2 4 2 1 1 1 2 1 1 A adjA A
- 48. Matrices - Operations The result can be checked using AA-1 = A-1 A = I The determinant of a matrix must not be zero for the inverse to exist as there will not be a solution Nonsingular matrices have non-zero determinants Singular matrices have zero determinants
- 49. Simple 2 x 2 case So that for a 2 x 2 matrix the inverse can be constructed in a simple fashion as a c b d A A a A c A b A d 1 •Exchange elements of main diagonal •Change sign in elements off main diagonal •Divide resulting matrix by the determinant z y x w A 1
- 50. Simple 2 x 2 case Example 2 . 0 4 . 0 3 . 0 1 . 0 2 4 3 1 10 1 1 4 3 2 1 A A Check inverse A-1 A=I I 1 0 0 1 1 4 3 2 2 4 3 1 10 1
- 51. Ch5_51 5.1 Eigenvalues and Eigenvectors Definition Let A be an n n matrix. A scalar is called an eigenvalue of A if there exists a nonzero vector x in Rn such that Ax = x. The vector x is called an eigenvector corresponding to . Figure 5.1
- 52. Ch5_52 Computation of Eigenvalues and Eigenvectors Let A be an n n matrix with eigenvalue and corresponding eigenvector x. Thus Ax = x. This equation may be written Ax – x = 0 given (A – In)x = 0 Solving the equation |A – In| = 0 for leads to all the eigenvalues of A. On expending the determinant |A – In|, we get a polynomial in . This polynomial is called the characteristic polynomial of A. The equation |A – In| = 0 is called the characteristic equation of A.
- 53. Find the eigenvalues and eigenvectors of the matrix 5 3 6 4 A Let us first derive the characteristic polynomial of A. We get Solution 5 3 6 4 1 0 0 1 5 3 6 4 2 I A 2 18 ) 5 )( 4 ( 2 2 I A We now solve the characteristic equation of A. The eigenvalues of A are 2 and –1. The corresponding eigenvectors are found by using these values of in the equation(A – I2)x = 0. There are many eigenvectors corresponding to each eigenvalue. 1 or 2 0 ) 1 )( 2 ( 0 2 2
- 54. • For = 2 We solve the equation (A – 2I2)x = 0 for x. The matrix (A – 2I2) is obtained by subtracting 2 from the diagonal elements of A. We get 0 2 1 3 3 6 6 x x This leads to the system of equations giving x1 = –x2. The solutions to this system of equations are x1 = –r, x2 = r, where r is a scalar. Thus the eigenvectors of A corresponding to = 2 are nonzero vectors of the form 0 3 3 0 6 6 2 1 2 1 x x x x 1 1 2 2 1 1 v 1 1 x x r x
- 55. • For = –1 We solve the equation (A + 1I2)x = 0 for x. The matrix (A + 1I2) is obtained by adding 1 to the diagonal elements of A. We get 0 2 1 6 3 6 3 x x This leads to the system of equations Thus x1 = –2x2. The solutions to this system of equations are x1 = –2s and x2 = s, where s is a scalar. Thus the eigenvectors of A corresponding to = –1 are nonzero vectors of the form 0 6 3 0 6 3 2 1 2 1 x x x x 1 2 2 2 2 2 v 1 1 x x s x
- 56. Find the eigenvalues and eigenvectors of the matrix 2 2 2 2 5 4 2 4 5 A The matrix A – I3 is obtained by subtracting from the diagonal elements of A.Thus Solution 2 2 2 2 5 4 2 4 5 3 I A 2 2 2 2 5 4 0 1 1 2 2 2 2 5 4 2 4 5 3 I A The characteristic polynomial of A is |A – I3|. Using row and column operations to simplify determinants, we get
- 57. 2 2 ) 1 )( 10 ( ) 1 )( 10 )( 1 ( ] 10 11 )[ 1 ( ] 8 ) 2 )( 9 )[( 1 ( 2 4 2 2 9 4 0 0 1 We now solving the characteristic equation of A: The eigenvalues of A are 10 and 1. The corresponding eigenvectors are found by using three values of in the equation (A – I3)x = 0. 10 , 1 0 ) 1 )( 10 ( 2
- 58. Matrices and Linear Equations Example 3 2 1 2 2 3 3 2 1 2 1 3 2 1 x x x x x x x x The equations can be expressed as 3 1 2 1 2 1 0 1 2 1 1 3 3 2 1 x x x
- 59. Linear Equations When A-1 is computed the equation becomes 7 3 2 3 1 2 5 . 2 5 . 3 5 . 1 0 . 1 0 . 2 0 . 1 5 . 0 5 . 0 5 . 0 1 B A X Therefore 7 , 3 , 2 3 2 1 x x x
- 60. RANK OF A MATRIX Let A be any m n matrix. Then A consists of n column vectors a₁, a₂ ,....,a, which are m-vectors. DEFINTION: The rank of A is the maximal number of linearly independent column vectors in A, i.e. the maximal number of linearly independent vectors among {a₁, a₂,......, a}. If A = 0, then the rank of A is 0. We write rk(A) for the rank of A. Note that we may compute the rank of any matrix-square or not
- 61. Let us see how to compute 22 matrix: EXAMPLE : rk ( A) 2 if det A ad bc 0,since both columnvectors are independent in this case. The rank of a 22 matrix A = a b is given by c d rk(A) = 1 if det(A) = 0 but A 0 = 0 0 ,since both column vectors 0 0 RANK OF 22 MATRIX are not linearly independent, but there is a single column vector that is linearly independent (i.e. non-zero). rk(A) = 0 if A = 0 How do we compute rk(A) of m x n matrix?
- 62. EXAMPLE Gauss elimination: * Find the rank of a matrix 0 2 2 4 2 2 1 1 A= 0 2 0 1
- 63. SOLUTION: We use elementary row operations: 2 4 2 4 1 0 2 1 1 0 2 1 A= 0 2 0 2 0 2 2 1 0 0 2 1 Since the echelon form has pivots in the first three columns, A has rank, rk(A) = 3. The first three columns of A are linearly independent.
- 64. EXAMPLE: Find the rank of a matrix using normalform, 2 3 4 5 3 4 5 5 6 10 11 6 A= 4 7 9 12 Solution: Reduce the matrix to echelon form, 2 3 4 5 1 0 0 0 4 5 1 2 5 6 0 0 10 11 0 0 6 0 3 4 7 0 3 0 9 12 0 0 rank(A)=2