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COMM 161

COMM 161. Intro to Mathematical Analysis for Management . Agenda. Section 1: Matrices/Linear Equations (15 min) Section 2: Matrix Algebra/Applications (15 min) Section 3: Input Output Models (15 min) Section 4 : Probability and Markov Systems(20 min)

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COMM 161

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  1. COMM 161 Intro to Mathematical Analysis for Management

  2. Agenda • Section 1: Matrices/Linear Equations (15 min) • Section 2: Matrix Algebra/Applications (15 min) • Section 3: Input Output Models (15 min) • Section 4: Probability and Markov Systems(20 min) • Section 5: Rates of Change & Derivatives (20 min) • Section 6: Techniques of Differentiation (20 min) • Section 7: Applications of Derivatives (20 min) • Section 8: Functions of Several Variables (20 min) • Section 9: 3D Optimization (20 min)

  3. COMM 161 Section 1: Systems of Linear Equations and Matrices

  4. Section 1: Systems of Linear Equations and Matrices • Three Ways of Solving a System of Equations: • Graphically • Algebraically via Substitution • Algebraically via Elimination • We will use the theory of matrices to solve systems of equations

  5. Section 1: Systems of Linear Equations and Matrices Coefficients Constants • A Matrix is a rectangular array of numbers • System: • 3x-y = 2 • x+y = 10

  6. Section 1: Systems of Linear Equations and Matrices • Different Row Operations for Solving a System: • Interchange any two equations

  7. Section 1: Systems of Linear Equations and Matrices • Different Row Operations for Solving a System: • Multiply a row by any non-zero constant

  8. Section 1: Systems of Linear Equations and Matrices • Different Row Operations for Solving a System: • Replace an equation by the sum of that equation and a constant multiple of any other equation

  9. Section 1: Systems of Linear Equations and Matrices • Reduced Row Echelon Form • The first non-zero entry in each row is one • The columns of the pivots are clear • Pivoting – Using a coefficient to clear a column • There is only one ‘one’ per row

  10. Section 1: Systems of Linear Equations and Matrices • Guass-Jordan Elimination Method • Write the augmented matrix • Interchange rows, if necessary, to obtain a nonzero first entry. Pivot on this entry. • Interchange rows, if necessary, to obtain a nonzero second entry in the second row. Pivot on this entry. • Continue until in reduced row form.

  11. Section 1: Systems of Linear Equations and Matrices • Possible Outcomes for a system: • Unique Solution • No Solution • Infinitely Many Solutions Reduced Row Form Final Row doesn’t make sense, 0+0+0 does not equal 1 The last row is true in all cases, therefore there is an infinite amount of solutions

  12. Section 1: Systems of Linear Equations and Matrices • Applying this information: • The lazy scholar makes its cheesecakes using three different types of cream cheese: A, B, C. The chocolate cake uses 1 pound of A and 2 pounds of B. The vanilla cake uses 2 pounds of A and 3 pounds of C. The strawberry cake uses 3 pounds of B and 4 pounds of C. Lazy scholar has a massive abundance of cream cheese and wants to use 34 pounds of A, 38 pounds of B and 60 pounds of C, how many cheesecakes can be made?

  13. Section 1: Systems of Linear Equations and Matrices

  14. Section 1: Systems of Linear Equations and Matrices Lazy Scholar can make 10 chocolate cheesecakes, 12 vanilla cheesecakes and 6 strawberry cheesecakes.

  15. COMM 161 Section 2: Matrix Algebra and Applications

  16. Section 2: Matrix Algebra and Applications • Matrix Dimensions • A matrix is an ordered rectangular array of real numbers with m rows and n columns. • Matrix Entries • Denoted by aij • ithrow, and jth column • a14 = 0 or a31 = -2 So, this is an example of a 3x4 matrix or 3 rows by 4 columns

  17. Section 2: Matrix Algebra and Applications • Addition and Subtraction of Matrices • Matrices must be the same size • The sum of A+B is found by adding corresponding entries in the two matrices • The difference of A-B is found by subtracting the corresponding entries in B and A

  18. Section 2: Matrix Algebra and Applications

  19. Section 2: Matrix Algebra and Applications • Scalar Multiplication • Any matrix A can be multiplied by c, which is a real number

  20. Section 2: Matrix Algebra and Applications • Transpose of a Matrix • Basically, you flip the rows with the columns and the columns with the rows.

  21. Section 2: Matrix Algebra and Applications • Matrix Multiplication – Size of the Product • If A is a matrix of size m x n and B is a matrix of size n x p then the product of AB will be a matrix of size m x p • I.E. Matrix A has a size of 3x2 • Matrix B has a size of 2x5 • The product of Matrix AB will be 3x5 • Note, The column size of A must equal the row size of B • Matrix A Size 3x4 and Matrix B 3x4 • The Product cannot be computed

  22. Section 2: Matrix Algebra and Applications For matrices A,B, and C, let AB = C. Then And so on…

  23. Section 2: Matrix Algebra and Applications • The identity matrix of size n is given by: Diagonal of 1’s, zeros every other entry

  24. Section 2: Matrix Algebra and Applications I is the identity matrix, O is the zero matrix. For matrices A, B, and C, the following hold: Associative law: (AB)C = A(BC) (A + B) + C = A + (B + C) Distributive law: A(B + C) = AB + AC (A+B)C = AC + BC Zero matrix: OA = AO = O Identity matrix: IA = AI = A

  25. Section 2: Matrix Algebra and Applications • Finding the inverse of a matrix • Given the square matrix A. • Adjoin the identity matrix I (of the same size) to form the augmented matrix: [A | I] • Use row operations to reduce the matrix to the form: [I | B] (if possible) • Matrix B is the inverse of A Ex. Find the inverse of A

  26. Section 2: Matrix Algebra and Applications Step 1 Step 2 Step 3

  27. Section 2: Matrix Algebra and Applications Step 4 Step 5 Step 6

  28. Section 2: Matrix Algebra and Applications Step 7

  29. Section 2: Matrix Algebra and Applications D is called a determinant. If D = 0 then the matrix is singular (has no inverse). Ex.

  30. Section 2: Matrix Algebra and Applications

  31. Section 2: Matrix Algebra and Applications • If AX = B is a linear system of equations (number of equations = number of variables) and A-1 exists, then X = A-1B is the unique solution of the system. AX = B A-1AX = A-1B (A-1A=I) IX = A-1B Finally, X = A-1B AX = B

  32. Section 2: Matrix Algebra and Applications Multiply by the inverse So (–2 , 3) is the solution

  33. COMM 161 Section 3: Input-Output Models

  34. Section 3: Input-Output Models In an input – output model, the matrix equation giving the net output of goods and services needed to satisfy external demand is: External Demand Production vector Technology Matrix The solution to the equation is: Which gives the amount of goods and services that must be produced to satisfy external demand.

  35. Section 3: Input-Output Models Ex. A simple economy consists of manufacturing (M) and service (S) and has the following input-output model (units are in millions of dollars) : M S M S • Determine the amount of manufactured products consumed for $300 million worth of service. (0.1)(300) = 30 So $30 million manufactured products are consumed

  36. Section 3: Input-Output Models • Find the gross output of goods needed to satisfy a consumer demand for $90 million worth of manufactured products and $60 million worth of service. $185.4 million worth of manufactured products and $212.7 million worth of service.

  37. Section 3: Input-Output Models • If A is the technology matrix, then the ijth entry of (I – A) –1 is the change in the number of units Sector i must produce in order to meet a one-unit increase in external demand for Sector j products. Change in production = (I – A) –1D+ Where D+ is the change in external demand.

  38. Post-midterm • I apologize in advance for the boring slides • Tips when practicing: • Communication and thinking • Applications! • General optimization • Buckle down, let’s start

  39. Probability and Markov Systems Some Key Terms • Sample space -Disjoint • Event (Mutually exclusive) • Complement -Probability distribution • Union -Relative frequency • Intersection -Trials/sample size

  40. Probability and Markov Systems Theoretical Probability P(E) = Number of favorable outcomes = n(E) Total number of outcomes n(S)

  41. Probability Rules • All probabilities are between 0 and 1 inclusive 0 <= P(E) <= 1 • The sum of all the probabilities in the sample space is 1 • The probability of an event which cannot occur is 0. • The probability of any event which is not in the sample space is zero. • The probability of an event which must occur is 1. • The probability of the sample space is 1. • The probability of an event not occurring is one minus the probability of it occurring. P(E') = 1 - P(E)

  42. Addition rule -General -Mutually exclusive events Conditional probability -Independent events -Dependent events

  43. Markov Systems -States -Time-step -State transition diagram -Transition matrix -State transition probabilities -Initial distribution vector -Steady-state vector

  44. Average Rate of Change Introduction to Derivatives -Average rate of change -Instantaneous rate of change -Differentiation -Tangent line

  45. Techniques of Differentiation -Product rule -Quotient rule -Chain rule -Logarithmic and exponential functions -Implicit differentiation

  46. Quick Tips for Differentiation • First derivative • Rate of change, velocity, marginal revenue • Second derivative • Acceleration, change in the ___ rate • Revenues = Price x Quantity • Profit = Revenues – Cost • Formula sheet! Use chapter summaries

  47. Applications of the Derivative • Relative extrema • Maxima and minima • Absolute extrema • Maximum and minimum • Critical points • Stationary (maximum or minimum) • Endpoints • Singular • Fence, volume

  48. Concavity • Up or down • Point of inflection • Point of diminishing return • Optimization question • Find the first derivative • Critical points • First derivative test • Solve for optimized value • Elasticity

  49. Inventory Management DON’T PANIC

  50. Functions of Several Variables • Distance formula • Graph of a function of several variables • Paraboloid • Level curves • Saddle point • Partial derivatives • Marginal costs • Mixed partial derivative

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