Finite Math with Applications
Finite Math with Applications MATH 1313
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This 12 page Class Notes was uploaded by Alvena McDermott on Saturday September 19, 2015. The Class Notes belongs to MATH 1313 at University of Houston taught by Staff in Fall. Since its upload, it has received 20 views. For similar materials see /class/208424/math-1313-university-of-houston in Mathmatics at University of Houston.
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Date Created: 09/19/15
Section 23 Solving Systems of Linear Equations 11 In the previous section we studied systems with unique solutions In this section we will study systems of linear equations that have in nitely many solutions and those that have no solution We also will study systems in which the number of variables is not equal to the number of equations in the systemi A System of Equations with an In nite Number of Solutions Example 1 The following augmented matrix in rowreduced form is equivalent to the augmented matrix of a certain system of linear equations Use this result to solve the system of equations 1 0 1 3 0 1 5 2 0 0 0 0 A System of Equations That Has No Solution Example Given the following system xyz1 3x y z4 x5y52 1 In using the GaussJordan elimination method the following equivalent matrix was obtained note this matrix is not in rowreduced form let s see why 1 1 1 1 0 4 4 1 0 0 0 1 Look at the last row It reads 0x 0y Oz 1 in other words 0 1 This is never true So the system is inconsistent and has no solution Systems with No Solution If there is a row in the augmented matrix containing all zeros to the left of the vertical line and a nonzero entry to the right of the line then the system of equations has no solution Section 213 Solving Systems of Linear Equations II Theorem I If the number of equations is greater than or equal to the number of variables in a linear system then one of the following is true at The system has no solution b The system has exactly one solution or The system has in nitely many solutions 11 If there are fewer equations than variables in a linear system then the system either has no solution or it has in nitely many solutions Example 2 Solve the system of linear equations using the GaussJordan elimination method x2y 3z 2 3x y 22l 2x3y 52 3 Section 23 Solving Systems of Linear Equations 11 Example 3 Solve the system of linear equations using the GaussJordan elimination method 2x3y2 x3y 2 x y3 Section 23 Solving Systems of Linear Equations 11 Example 4 Solve the system of linear equations using the GaussJordan elimination method x 2y2 7x l4yl4 3x 6y6 Example 5 Solve the system of linear equations using the GaussJordan elimination method 4xy z4 8x2y 228 Section 23 Solving Systems of Linear Equations 11 Section 24 Matrices A matrix is an ordered rectangular array of numbers A matrix with m rows and n columns has size or dimension m x n The entry in the ith row and jth column is denoted by 111 A matrix with only one column or one row is called a column matrix or column vector or row matrix or row vector respectively The real numbers that make up the matrix are called entries or elements of the matrix 2 7 7 5 3 9 Example 1 Gwen A 0 10 20 1 3 11 a What is the dimension of A b identify 143 A square matrix is a matrix having the same number of rows as columns E 1 3 9 Xam e p 4 1 Equality of Matrices Two matrices are equal if they have the same dimension and their corresponding entries are equal Example 2 Solve the following matrix equation for W x y and z 10w 8 51 10 w 8 8 5x 0 5 3zll 11 7 7y 12 5 9z3 11 7 Section 24 Matrices l Addition and Subtraction of Matrices If A and B are two matrices of the same dimension 1 The sum A B is the matrix obtained by adding the corresponding entries in the two matrices 2 The difference A B is the matrix obtained by subtracting the corresponding entries in B from Al Laws for Matrix Addition If A B and C are matrices of the same dimension then 1 A B B A 2 ABC ABC Example 3 Refer to the following matrices 8 3 1 5 4 1 10 8 3 4 1 3 A0 9 4B 8 4 8 C andD i 5 4 2 8 5 1 9 6 7 10 15 2 prossible at computeA B b compute B D c compute D Cl Section 24 Matrices Transpose of a Matrix If A is an m x n matrix with elements a I then the transpose of A is the n x m matrix ATWltl l elements a Example 4 Given the following matrices nd their transpose 3 0 6 a B 10 100 3 4 11 3 Scalar Multiplication A matrix A may be multiplied by a real number called a scalar in the context of matrix algebra Scalar Product If A is a matrix and c is a real number then the scalar product cA is the matrix obtained by multiplying each entry of A by c The zero matrix is one in which all entries are zero Note 0 represents the zero matrix Section 24 Matrices 3 1 2 1 4 1 2 3 i Example 5 Let A B and C nd 1fp0551b1e 3 4 7 9 6 9 1 at 2B A b BC C ZX 3A B Example 6 Solve for the variables in the matrix equation 1 2 14 6 25 3 8 4 3 9 y 13 21 v 8 6 o x 4 w1 w Section 24 Matrices M 1313 Section 25 1 Multiplication of Matrices If A is a matrix of size m x n and B is a matrix of size n x p then the product AB is de ned and is a matrix of size m x p So two matrices can be multiplied if and only if the number of columns in the rst matrix is equal to the number of rows in the second matrixi Example 1 Multiple the given matrices 6 l 2 3isa1x3matrix 5 isa3x1matrix 4 When multiplied the ending matrix will be 1 x 1 6 1235 4 Here is how you multiply all 312 1911 all X b11 a12 X b21 a21 a22 b21 a21 X b11 a22 X b21 Example 2 Multiply the given matrices 2 1 1 3 0 2 1 1 2 0 1 2 0 1 0 0 1 M 1313 Section 25 Example 3 Mike and Sam have stock as follows BAC GM IBM TRW A 200 300 100 200 Mike is this row and Sam row three 100 200 400 0 At the close of trading on a certain day the price share is BAC 54 B GM 48 IBM 98 TRW 82 AB Example 4 Multiply the following matrices if possible 3 9 3 l 4 1 3 0 10 9 l LetA B 2 0 3 C andD 2 4 l 6 4 9 l 2 l 4 compute if possible AB CA M 1313 Section 25 3 Laws for Matrix Multiplication If the products and sums are de ned for the matrices A B and C then 1 ABC ABC 2i AB C AB AC Note In general matrix multiplication is not commutative that is AB BA Example 5 If A and B are matrices we will look at the product AB and BA 3 4 l 2 A B 2 0 5 7 AB BA You can not divide matrices We use the inverse In 26 you will learn how to nd and inverse In this section we will give you some background a i If a is a nonzero real number then there exists a unique real number such that 21 12121 Let s look at matrices 2 1 M 1 2 M 1 l 3 4 MoM4 M 1313 Section 25 4 Identity Matrix The square matrix of size n having ls along the main diagonal and zeros elsewhere is called the identity matrix of size n 1 0 0 0 0 0 0 1 0 0 0 0 i i i 0 0 1 0 0 0 The identity matrix of s1ze n is given by I1 0 0 1 HA is a square matrix of size n then InA AI1 A 1 2 4 Example 6 Let A 3 5 X x1 and B 12 Show that the equation z AX B represents a linear system of two equations in two unknowns Matrix Representation A system of linear equations may be written in a compact form with the help of matrices Example 7 Given the following system of equations write it in matrix form 2x 4yz6 3x6y 52 1 x 3y7z0
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