![Matrices - Introduction
A matrix is denoted by a bold capital letter and the elements
within the matrix are denoted by lower case letters
e.g. matrix [A] with elements aij
mn
ij
m
m
n
ij
in
ij
a
a
a
a
a
a
a
a
a
a
a
a
2
1
2
22
21
12
11
...
...
i goes from 1 to m
j goes from 1 to n
Amxn=
mAn](https://image.slidesharecdn.com/matricesanddeterminants-240829055232-1bdae02a/85/Matrices-and-Determinants-ppt-5-320.jpg)
![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.](https://image.slidesharecdn.com/matricesanddeterminants-240829055232-1bdae02a/85/Matrices-and-Determinants-ppt-43-320.jpg)
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- 1. Math 11: Linear Algebra Topic: Matrices
- 3. 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
- 4. 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
- 5. Matrices - Introduction A matrix is denoted by a bold capital letter and the elements within the matrix are denoted by lower case letters e.g. matrix [A] with elements aij mn ij m m n ij in ij a a a a a a a a a a a a 2 1 2 22 21 12 11 ... ... i goes from 1 to m j goes from 1 to n Amxn= mAn
- 6. 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
- 7. 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
- 8. 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
- 9. 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
- 10. 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
- 11. 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 a = 1 for some or all i = j ij ij a a 0 0
- 12. 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
- 13. 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
- 14. 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
- 15. 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
- 16. 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
- 18. 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
- 19. 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
- 20. 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
- 21. Matrices - Operations Commutative Law: A + B = B + A Associative Law: A + (B + C) = (A + B) + C = A + B + C 9 7 2 5 8 8 3 2 4 6 5 1 6 5 2 1 3 7 A 2x3 B 2x3 C 2x3
- 22. Matrices - Operations A + 0 = 0 + A = A A + (-A) = 0 (where A is the matrix composed of aij as elements) 1 2 2 2 2 5 8 0 1 0 2 1 7 2 3 2 4 6
- 23. 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
- 24. 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
- 25. 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)
- 26. Matrices - Operations B x A = Not possible! (2x1) (4x2) A x B = Not possible! (6x2) (6x3) Example A x B = C (2x3) (3x2) (2x2)
- 27. 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
- 28. 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
- 29. 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
- 30. Matrices - Operations AB not generally equal to BA, BA may not be conformable 0 10 6 23 0 5 2 1 2 0 4 3 20 15 8 3 2 0 4 3 0 5 2 1 2 0 4 3 0 5 2 1 ST TS S T
- 31. Matrices - Operations If AB = 0, neither A nor B necessarily = 0 0 0 0 0 3 2 3 2 0 0 1 1
- 32. Matrices - Operations TRANSPOSE OF A MATRIX If : 1 3 5 7 4 2 3 2 A A 2x3 1 7 3 4 5 2 3 2 T T A A Then transpose of A, denoted AT is: T ji ij a a For all i and j
- 33. 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 2 A A 1 7 3 4 5 2 2 3 T T A A 2 x 3 3 x 2
- 34. 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
- 35. Matrices - Operations 1. (A+B)T = AT + BT 9 7 2 5 8 8 3 2 4 6 5 1 6 5 2 1 3 7 9 5 7 8 2 8 9 5 7 8 2 8 3 6 2 5 4 1 6 1 5 3 2 7
- 36. Matrices - Operations (AB)T = BT AT 8 2 3 0 2 1 0 1 2 1 1 8 2 8 2 2 1 1 3 2 0 0 1 1
- 37. 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
- 38. Matrices - Operations When the original matrix is square, transposition does not affect the elements of the main diagonal d b c a A d c b a A T The identity matrix, I, a diagonal matrix D, and a scalar matrix, K, are equal to their transpose since the diagonal is unaffected.
- 39. Matrices - Operations INVERSE OF A MATRIX Consider a scalar k. The inverse is the reciprocal or division of 1 by the scalar. Example: k=7 the inverse of k or k-1 = 1/k = 1/7 Division of matrices is not defined since there may be AB = AC while B = C Instead matrix inversion is used. The inverse of a square matrix, A, if it exists, is the unique matrix A-1 where: AA-1 = A-1 A = I
- 41. Matrices - Operations Properties of the inverse: 1 1 1 1 1 1 1 1 1 1 ) ( ) ( ) ( ) ( ) ( A k kA A A A A A B AB T T A square matrix that has an inverse is called a nonsingular matrix A matrix that does not have an inverse is called a singular matrix Square matrices have inverses except when the determinant is zero When the determinant of a matrix is zero the matrix is singular
- 42. 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
- 43. 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.
- 44. 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.
- 45. 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
- 46. 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
- 47. 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
- 48. 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)
- 49. 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
- 50. Matrices - Operations Example 1: 2 1 1 3 A 5 ) 1 )( 1 ( ) 2 )( 3 ( A
- 51. 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
- 52. 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
- 53. 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
- 54. 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
- 55. 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
- 56. 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 ) (
- 57. 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
- 58. 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
- 59. 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
- 60. 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
- 61. 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
- 62. Matrix Inversion Simple 2 x 2 case
- 63. Simple 2 x 2 case Let d c b a A and z y x w A 1 Since it is known that A A-1 = I then 1 0 0 1 z y x w d c b a
- 64. Simple 2 x 2 case Multiplying gives 1 0 0 1 dz cx dy cw bz ax by aw bc ad A It can simply be shown that
- 65. Simple 2 x 2 case thus A d bc da d w d cw b aw d cw y b aw y 1 1
- 66. Simple 2 x 2 case A b bc da b x d cx b ax d cx z b ax z 1 1
- 67. Simple 2 x 2 case A c cb ad c y c dy a by c dy w a by w 1 1
- 68. Simple 2 x 2 case A a bc ad a z c dz a bz c dz x a bz x 1 1
- 69. 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
- 70. 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
- 71. Matrices and Linear Equations Linear Equations
- 72. Linear Equations Linear equations are common and important for survey problems Matrices can be used to express these linear equations and aid in the computation of unknown values Example n equations in n unknowns, the aij are numerical coefficients, the bi are constants and the xj are unknowns n n nn n n n n n n b x a x a x a b x a x a x a b x a x a x a 2 2 1 1 2 2 2 22 1 21 1 1 2 12 1 11
- 73. Linear Equations The equations may be expressed in the form AX = B where , , 2 1 1 1 2 22 21 1 12 11 n nn n n n n x x x X a a a a a a a a a A and n b b b B 2 1 n x n n x 1 n x 1 Number of unknowns = number of equations = n
- 74. Linear Equations If the determinant is nonzero, the equation can be solved to produce n numerical values for x that satisfy all the simultaneous equations To solve, premultiply both sides of the equation by A-1 which exists because |A| = 0 A-1 AX = A-1 B Now since A-1 A = I We get X = A-1 B So if the inverse of the coefficient matrix is found, the unknowns, X would be determined
- 75. 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
- 76. 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
- 77. Linear Equations The values for the unknowns should be checked by substitution back into the initial equations 3 2 1 2 2 3 3 2 1 2 1 3 2 1 x x x x x x x x 3 ) 7 ( ) 3 ( 2 ) 2 ( 1 ) 3 ( ) 2 ( 2 2 ) 7 ( ) 3 ( ) 2 ( 3 7 , 3 , 2 3 2 1 x x x