Section 6.1 Systems of Linear Equations Chapter 6 Section 6.1 Systems of Linear Equations

Section 6.1 Systems of Linear Equations Linear equations (1st degree equations) System of linear equations Solution of the system System of 2 linear equations in 2 variables: independent, dependent, inconsistent systems Solution methods: substitution and elimination. Example 1 (p. 313), 2, 4 (p.314), 6 (p. 316)

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6.2 Larger system of linear equations Two systems are equivalent if they have the same solutions. Elementary operations (to produce an equivalent system): Interchange any two equations Multiply both sides of an equation by a non-zero constant. Replace an equation by the sum of itself and a constant multiple of another equation in the system.

Elimination method Example 1 (p. 320) Elimination method for solving larger system of linear equations: Make the leading coefficient of the first equation 1. Eliminate the leading variable of the first equation from each later equation. Repeat steps 1 and 2 for the second equation. Repeat steps 1 and 2 for the third, fourth equation and so on, till the last equation. Then solve the resulting system by back substitution.

MATRIX METHODS Matrix Row, Column, Element (entry) Augmented matrix Row operations on matrices: Interchange any two rows. Multiply each element of a row by a non-zero constant. Replace a row by the sum of itself and a constant multiple of another row of the matrix. Example 2 (p. 322)

MATRIX METHODS Row echelon form: All rows having entirely zeros (if any) are at the bottom The first nonzero entry in each row is 1 (called leading 1). Each leading 1 appears to the right of the leading 1’s in any preceding rows. Example:

DEPENDENT AND INCONSISTENT SYSTEMS Example 9: Solution: The system has infinitely many solutions (the system is dependent) Example 11: Solution: the system has no solution (it is inconsistent)

GAUSS-JORDAN METHOD Example 1: (The system is independent)

GAUSS-JORDAN METHOD Example 2: (The system is inconsistent)

GAUSS-JORDAN METHOD Example 3: (The system is dependent)

GAUSS-JORDAN METHOD A matrix is said to be in reduced row echelon form if it is in row echelon form and every column containing a leading 1 has zeros in all its other entries. Example:

GAUSS-JORDAN METHOD Arrange the equations with the variables terms in the same order on the left of the equal sign and the constants on the right. Write the augmented matrix of the system. Use the row operations to transform the augmented matrix into reduced row echelon form: Stop the process in step 3 if you obtain a row whose elements are all zeros except the last one. In that case, the system is inconsistent and has no solutions. Otherwise, finish step 3 and read the solutions of the system from the final matrix.

6.3 Applications of Systems of Linear Equations Example 1: (p. 333) A company plans to spend $3 million on 200 new vehicles. Each van will cost $10000, each small truck $15000, and each large truck $25000. Past experience shows that the company needs twice as many vans as small trucks. How many of each kind of vehicles can the company buy?

6.3 Applications of Systems of Linear Equations Example 2: (p. 334) Ellen plans to invest a total of $100000 in a money market account, a bond fund, an international stock fund, and a domestic stock fund. She wants 60% of her investment to be conservative (money market and bonds). She wants the amount in international stocks to be one-forth of the amount in domestic stocks. Finally, she needs an annual return of $4000. Assuming she gets annual return of 2.5% on the money market account, 3.5% on the bond fund, 5% on the international stock fund, and 6% on the domestic stock fund, how much should she put in each investment?

6.3 Applications of Systems of Linear Equations Example 3: An animal feed is to be made from corn, soybean, and cottonseed. Determine how many units of each ingredient are needed to make a feed that supplies 1800 units of fiber, 2800 units of fat, and 2200 units of protein, given the information below: Corn Soybean Cottonseed Totals Fiber Fat Protein 10 30 20 40 25 1800 2800 2200

6.3 Applications of Systems of Linear Equations Example 4: The concentrations (in parts per million) of carbon dioxide (a greenhouse gas) have been measured at Mauna Loa, Hawaii, since 1959. The concentrations are known to have increased quadratically. The following table lists readings for 3 years: Use the given data to construct a quadratic function that gives the concentration in year x Use this model to estimate the carbon dioxide concentrations in 2010 and 2014. Year 1964 1984 2004 Carbon Dioxide 319 344 377

6.3 Applications of Systems of Linear Equations Example 5: Kelly Karpet Kleaners sells rug-cleaning machines. The EZ model weighs 10 pounds and comes in a 10-cubic-foot box. The compact model weighs 20 pounds and comes in an 8-cubic-foot box. The commercial model weighs 60 pounds and comes in a 28-cubic-foot box. Each of Kelly’s delivery van has 248 cubic feet of space and can hold a maximum of 440 pounds. In order for a van to be fully loaded, how many of each model should it carry?

6.4 Basic Matrix Operations Size of a matrix Row matrix Column matrix Square matrix Element of matrix A: aij : element in row i and column j

Sum of two matrices Sum of two matrices of the same size: Given matrices X and Y (both have the same size m  n). Matrix Z = X + Y has elements zij = xij + yij, where xij , yij, zij are the elements on the i-th row, j-th column of matrices X, Y and Z.

Additive inverse of a matrix A is the matrix –A in which each element is the additive inverse of the corresponding element of A. Zero matrix O: all elements are zeros. Identity property: A + O = O + A = A, A is any matrix.

Subtraction: The difference of X and Y (same size) is matrix Z, in which each element is the difference of the corresponding elements of X and Y, or, equivalently: Z = X – Y = X + (– Y)

Product of a scalar k and a matrix X is the matrix kX, each of whose elements is k times the corresponding element of X. Exercise: Let Find each of the following: 1. 2A 2. –3B 3. 3A – 10B

Product of a Row Matrix and a Column Matrix 6.5 MATRIX PRODUCT AND INVERSE Product of a Row Matrix and a Column Matrix

Matrix Product If A is an m × p matrix and B is a p × n matrix, then the matrix product of A and B, denoted AB, is an m × n matrix whose element in the ith row and jth column is the real number obtained from the product of the ith row of A and the jth column of B. If the number of columns in A does not equal the number of rows in B, then the matrix product AB is not defined.

Check Sizes Before Multiplication

MATRIX PRODUCT 7-1-67

Example

Product (Sigma Notation) Let A be an mn matrix and let B be an nk matrix. The product matrix AB (denoted C) is the mk matrix whose entry in the i-th row and j-th column is: Cij =

Properties Associative property: A(BC) = (AB)C, A+(B+C) = (A+B)+C Distributive property: A(B+C) = AB + AC Identity matrix I: On the main diagonal: all elements are 1 Elsewhere: all elements are 0 Not commutative: AB  BA in general

Definition of inverse matrix: Given matrix A, if exists matrix B so that AB = I, B is called inverse matrix, and denoted A-1 (read A-inverse). Singular, non-singular matrix Inverse matrix calculation: Form the augmented matrix [A| I] Perform row operations on [A| I] to get a matrix of the form [I | B]. Matrix B is A-1.

6.6 Applications of Matrices 1. Solving systems with matrices: System AX = B, where A is coefficient matrix, X is the matrix of variables, and B is the matrix of constants, is solved by first finding A-1. Then, if A-1 exists, X = A-1B. Example: 2x – 3y = 4 x + 5y = 2 Write matrices A, X, B in this example.

6.6 Applications of Matrices 2. Input-output analysis Input-output matrix A (or technological matrix) of an economy. Example 3.

6.6 Applications of Matrices 2. Input-output analysis Production matrix X Demand matrix D = X – AX Example 4.

6.6 Applications of Matrices 2. Input-output analysis In practice, A and D are known, we need to find the production matrix: X–1 = (I – A) –1D Example 6: An economy depends on 2 basic products: wheat and oil. To produce 1 ton of wheat requires .25 ton of wheat and .33 ton of oil. The production of 1 ton of oil consumes .08 ton of wheat and .11 ton of oil. Find the production that will satisfy the demand of 500 ton of wheat and 1000 ton of oil.