1. Everything you would want to know about the matrix and then some…
ENM 503 – The Matrix
Everything you would want to know about the matrix and then some…
this way

2. Applications Solving systems of linear equations Regression analysis
Markov processes
Linear programming
Nonlinear optimization
Queuing
Reliability
Inventory – MRP systems

3. Matrix and Vectors
A matrix is a rectangular array of elements which are operated on as a single object. The elements are often numbers but could be any mathematical object provided that it can be added and multiplied with acceptable properties.
Vectors are strongly related to matrices, they can be considered as a matrix having only a single row (row vector) or a single column (column vector).

4. Examples X is a 1 x 4 row vector, Y is a 3 x 1 column vector
A is a 3 x 3 matrix, and B is a 3 x 2 matrix

5. An m x n Matrix

6. Vector Matrix Operations
Vectors and matrices can be added (or subtracted) and multiplied when their dimensions are in agreement.
To add two vectors or two matrices having the same dimensions, just add their corresponding elements
A + B = {aij + bij}
To multiply two vectors, multiply corresponding elements and add. The result is a scalar (dot product). Both vectors must have the same number of elements.

7. Vector Example

8. Example Matrix Addition and scalar multiplication

9. Matrix Multiplication
If A is an m x n matrix and B is an n x p matrix, then C = A x B is an m x p matrix where
The i,j element of C is found by multiplying the ith row of A times the jth column of B (equivalent to a vector multiplication).

10. Example Matrix Multiplication

11. Properties of Matrix Operations
A (BC) = (AB) C
A (B+C) = AB + AC
(B+C) A = BA + CA
however A B  B A (both are defined only if A and B
are n x n matrices)
and A A = A2 (only if a square matrix, i.e dimension n x n)

12. An interesting sidelight
A · B = 0
does not necessarily imply that A = 0 or B = 0
For example:

13. The Transpose If A = {ajk} then At = {akj}
Each row of A becomes a column of At
If A = At, then A
is a symmetric matrix;
i.e. aij = aji

14. Properties of the Transpose
(At)t = A
(A + B)t = At + Bt
(kA)t = k At
(AB)t = Bt At
Quick student exercise:
Create an example to
illustrate each property
Quick student exercise: (AtA)t = AtA
Show that AtA is symmetric using the above properties

15. Some Special Matrices
The Identity Matrix (n x n) The Null Matrix

16. More Special Matrices
all zeros
Upper triangular Lower Triangular

17. The Diagonal Matrix
main diagonal

18. The Determinant
For a square (n x n) matrix A, the determinant is defined as a scalar computed from the sum of n! terms of the form (  a1i a2j … anr) ; the sign alternating and depending upon the permutation.

19. A 2 x 2 Determinant

20. A 3 x 3 determinant + -

21. Properties of Determinants
1. |A| = |At|
2. If ajk = 0 for all k or for all j, |A| = 0
3. Interchange any 2 rows, A': |A| = - |A'|
4. For scalar k, |kA| = k |A|
5. If there are 2 identical rows or columns, |A| = 0
6. |AB| = |A||B|
7. If A is triangular,

22. Cofactors
Minor – determinant of order (n-1) obtained by removing the jth row and kth column of A
Cofactor: (-1)j+k Minorjk = Ajk
Cofactor matrix - A matrix with elements that are the cofactors, term-by-term, of a given square matrix – [Ajk]
Adjoint Matrix = transpose of the cofactor matrix – [Ajk]t

23. Example Cofactor Matrix

24. The Adjoint Matrix
(adj-mat #2A((1 2 3)(0 4 5)(1 0 6)))  #2A(( )(5 3 -5)(-4 2 4))
(det #2A((1 2 3)(0 4 5)(1 0 6)))  22

25. Expansion by Cofactors (2 x 2)
Pierre-Simon Laplace
Born: 23 March 1749
in Normandy, France Died: 5 March 1827
in Paris, France

26. Expansion by Cofactors (3x3)

27. Example Expansion by Cofactors (3x3)
Expanding about row 1:

28. Matrix Inverse
A square matrix A may have an inverse matrix A-1 such that:
If such a matrix exists, then A is said to be nonsingular or invertible. The inverse matrix A-1 will be unique.
A square matrix A is said to be singular if |A| = 0.
If |A|  0, then A is said to be nonsingular

29. The Necessary Example

30. Finding A-1 for a 2 x 2 solve for x, y, z, and w in terms
of a, b, c, and d.

31. Properties of Inverses

32. Finding Inverses Method 1 – Adjoint Matrix
Method 2 – Gauss-Jordan Elimination Method
Elementary Row Operations (ERO)
define an augment matrix [A:I] where I is an n x n identity matrix
Perform ERO on [A:I] to obtain [I:A-1]
If |A| = 0, then A-1 does not exist!

33. Method 1: The Adjoint Method

34. Method 2 – The Gauss-Jordan Way
Carl Friedrich Gauss:  
Wilhelm Jordan

35. Elementary Row Operations (ERO)
Interchange ith and jth row:
Ri  Rj
Multiply the ith row by a nonzero scalar
Ri  kRi
Replace the ith row by k times the jth row plus the ith row
Ri  kRj + Ri

36. The Augmented Matrix
[ A : I ]
[ I : A-1]
ERO’s
need an example?

37. The Example

38. Matrices and Systems of Linear Equations
Ax = b

39. The Augmented Matrix Ax = b [ A : I : b ] [ I : A-1 : b’ ] x = b’
ERO’s

40. An Example
R1  R1 /3
R2  R2 – R1
R2  3 R2
R1  R1 – (5/3)R2

41. Solving systems of linear eqs. using the matrix inverse
Matrix solution:
AX = b
A-1 (AX) = (A-1A)X = IX =X = A-1b

42. Example System or Eqs

43. Cramer’s Rule for solving systems of linear equations
Given AX = b
Let Ai = matrix formed by
replacing the ith column with b,
then
Gabriel Cramer
Born: 31 July 1704 in Geneva, Switzerland Died: 4 Jan 1752 in Bagnols-sur-Cèze, France

44. An Example of Cramer’s Rule

45. Cramer solving a 3 x 3 system

46. Finding A-1 for a 2 x 2 solve for x, y, z, and w in terms
of a, b, c, and d.

47. Finding A-1 for a 2 x 2

48. Finding A-1 for a 2 x 2

49. Finding A-1 for a 2 x 2

50. A Numerical Example

51. Properties of Triangular Matrices
Triangular matrices have the following properties (prefix ``triangular'' with either ``upper'' or ``lower'' uniformly):
The inverse of a triangular matrix is a triangular matrix.
The product of two triangular matrices is a triangular matrix.
The determinant of a triangular matrix is the product of the diagonal elements.
A matrix which is simultaneously upper and lower triangular is diagonal
The transpose of a upper triangular matrix is a lower triangular matrix and vice versa

52. Determinants by Triangularization
R’2  -5/4 R1 + R2
R’3  -4 R2 + R3

53. Solving Systems of Equations the Easy Way
UseThe Genie!

54. Matrices and Determinants
1/2/2019 rd

55. Matrix
A matrix is a rectangular arrangement of entries, displayed in roes and columns. A square matrix has a determinant value used in solving systems of linear equations.
1/2/2019 rd

56. Principal Diagonal
The principal diagonal is a11, a22, a33, … amn in this m rows by n columns matrix.
The sum of the diagonal elements is called the trace. A matrix with all zeros is the null matrix.
A matrix with 1's along the main diagonal and zeros elsewhere is the identity matrix.
1/2/2019 rd

57. Transpose Matrix
The transpose of a matrix is the rows of the matrix become the columns of its transpose. A matrix A is symmetric if A = AT skew-symmetric if A = -AT orthogonal if A = (AT)-1 (AT)T = A and (rA)T = rAT where r is a scalar quantity. (A + B)T = AT + BT
1/2/2019 rd

58. Addition/Subtraction
Matrices are added and subtracted component-wise.
(M+ #2A((3 5 -2) (4 6 5) (8 9 7)) #2A(( ) (4 5 6) (7 8 9))) 
#2A((4 7 1)( )( ))
4 7 1
1/2/2019 rd

59. Multiply Matrix
AB = (dot-product '(1 0) '(3 1))  3 (dot-product '(1 0) '(-1 2))  -1 (dot-product '(1 2) '(3 1))  5 (dot-product '(1 2) '(-1 2))  3 (M* #2A((1 0)(1 2)) #2A((3 -1)(1 2)))  #2A((3 -1)(5 3))
1/2/2019 rd

60. Delivery
Glazed and jelly-filled donuts are delivered daily costing $10 a dozen for glazed and $12 a dozen for jelly-filled. On the first day 100 dozens of glazed and 120 dozen of jelly-filled were delivered; on the next day 90 dozen and 150 dozen and on the third day 112 dozen and 160 dozen respectively. Use matrices to find the total cost for each day.
1/2/2019 rd

61. Matrix Properties
(An)-1 = (A-1)n (inverse (expt-matrix matrix n)) (expt-matrix (inverse matrix) n)) (AT )-1 = (A-1)T (AB)T = BTAT (AB)-1 = B-1A-1 det A = 1 / det A-1 Determinants can be calculated using co-factors and minors. Only square matrices have determinants.
1/2/2019 rd

62. Inverse Matrix
AA-1 = I, the Identity matrix. Find the inverse matrix of #2A( (1 2) (3 8) ) 1a + 3b = 1 2a + 8b = 0 1c + 3d = 0 2c + 8d = 1 (solve '((1 3 1)(2 8 0)))  (4 -1) (solve '((1 3 0) (2 8 1)))  (-3/2 1/2)
(4 -1)
(-3/2 1/2)
1/2/2019 rd

63. Determinants
|A| = |AT| has determinant has determinant =
1/2/2019 rd

64. Co-factors
Evaluate (7 – 12) = 5 (det #2A ((1 2 3)(4 6 7)(0 1 0)))  5
1/2/2019 rd

65. ii
ei/2 = cos /2 + i sin /2 = i (ei/2)i = e-/2 = Euler's Equation states that ei + 1 = 0 => ei = -1 i = ln(-1) => i/2 = ln (-1)1/2 = ln i ii = ei ln i = (ei/2)i  and e are transcendental Using a fixed-size font, they can't be written out on a piece of paper as big as the universe.