More with Matrices and the Calculator Video LectureSections 11.1 and 11.2Course Learning Objectives:1)Solve systems of linear equations using technology.2)Perform the algebra of matrices, find inverses of matrices, and solve matrix equations.Weekly Learning Objectives:1)Solve systems of equations by elimination.2)Identify inconsistent systems of equations containing two variables.3)Express the solution of a system of dependent equations containing two variables.4)Solve systems of three equations containing three variables.5)Identify inconsistent systems of equations containing three variables.6)Express the solution of a system of dependent equations containing three variables.7)Perform row operations on a matrix.8)Solve a system of linear equations using matrices.page 1More Matrices and the CalculatorAn equation of the form is called LINEAR since its graph is aEB FC œ Gline. For a system of linear equations , exactly one of the following is true.1. The system has exactly one solution. Such a system is said to be2. The system has no solutions. Such a system is said to be3. The system has infinitely many solutions. Such a system is said to beThis is easy to see geometrically if we have a system in two variables.Although it is more difficult to visualize if we have a system of equations8with variables, the statement is still true.8In solving a system of equations we are attempting to find a solution bycreating an equivalent system whose solutions are more obvious for us tosee. An equivalent system is a system whose solutions are the same asthe original system. If we use the elimination method, then there are 3operations that can be used to create an equivalent system.1. Add a nonzero multiple of one equation to another.2. Multiply an equation by a nonzero constant.3. Interchange the position of two equations.page 2Solve each of the systems:1. B  $C œ )B %C œ "$2. B  $C œ )#B  'C œ "#3. B  $C œ )#B  'C œ "'page 3Note: A system is said to be in triangular form if the second equation doesnot contain the first variable, the third equation does not contain either thefirst or second variable, etc. The following is an example of a system in atriangular form. B  $C $D œ )C  &D œ "'D œ % It is easy to solve a system that is in triangular form by using backsubstitution. Our goal in this section is to change a system of linearequations into an equivalent system that is in triangular form.This process is referred to as Gaussian elimination.page 4We can express a system of equations as a rectangular array of numberscalled a MATRIX. Matrices (plural of matrix) provide us will an efficientmethod for solving a system of linear equations.A matrix is a rectangular array of rows and columns.7 ‚ 8 7 8 Ô ×Ö ÙÖ ÙÖ ÙÖ ÙÖ ÙÖ ÙÖ ÙÖ ÙÕ Ø+ + + Þ Þ Þ ++ + + Þ Þ Þ ++ + + Þ Þ Þ +Þ Þ Þ Þ Þ Þ ÞÞ Þ Þ Þ Þ Þ ÞÞ Þ Þ Þ Þ Þ Þ+ + + Þ Þ Þ +"" "# "$ "8#" ## #$ #8$" $# $$ $87" 7# 7$ 78The numbers are the entries of the matrix. The subscript on the entry+ +34 34indicates that it is the row and the column.3 4>2 >2We write a system of linear equations in matrix form by writing only thecoefficients and constants that appear in the equations. The resultingmatrix is called the for the system.augmented matrix B  $C $D œ % " $ $ %#B C  &D œ "" # " & ""B  #C D œ ! " # " !Ô ×Õ ØWe can perform the operations referred to in solving a system of equationson the matrix form of the equation. When performed on the matrix form ofan equation they are referred to as Elementary Row Operations andexpressed in the following way.1. Add a nonzero multiple of one row to another.2. Multiply a row by a nonzero constant.3. Interchange two rows.page 5To solve a system of linear equations using its augmented matrix we usethe elementary row operations to arrive at a matrix that is “triangular”. Thematrix is said to be in Row Echelon Form (REF). This method is oftenreferred to as Gaussian Elimination.Row Echelon Form:1. The first nonzero number in each row (reading from left toright) is 1. This is called the leading entry.2. The leading entry in each row is to the right of the leading entryin the row immediately above it.3. All rows consisting entirely of zeros are at the bottom of thematrix.The following matrix is in row echelon form.Ô ×Ö ÙÖ ÙÕ Ø"""# $ $! % #! ! "! ! ! !Reduced Row-Echelon Form (RREF):Also satisfies the following condition:4. Every number above and below each leading entry is a 0.The following matrix is in reduced row-echelon form. Ô ×Õ Ø"""! ! #! ! "! ! &This process is also referred to as Gauss-Jordan elimination.page 6Solve using a matrix: %B )C  %D œ %$B 'C &D œ  "$ #B C "#D œ  "(Find REF and RREFNow solve using your calculator:page 7A system which has no solution system is said to be inconsistent)Ðhas a reduced row echelon form similar to the following.rref: Î ÑÏ Ò"  $ # "#! " "  "!! ! ! "A system which has an infinite number of solutions (system said tobe dependent) has a reduced row echelon form similar to either ofthe following.rref: Assume variables are Î ÑÏ Ò" ! !  % !! " ! ! &! ! " $ !Bß Cß Dß AThe solution to the system corresponding to the augmented matrixabove is represented asrref: Assume variables are Î ÑÏ Ò" !  $  % !! " ! ! &! ! ! ! !Bß Cß Dß AThe solution to the system corresponding to the augmented matrixabove is represented aspage 85. 'B  C  D œ %  "#B #C #D œ  ) &B C  D œ $page 96. B  #C D œ ! #B #C  $D œ  $ C  D œ  "  B %C #D œ "$page 107. Carletta has $10,000 to invest. As her financial consultant, you recommend that she invest in Treasury bills that yield 6%, Treasury bonds that yield 7%, and corporate bonds that yield 8%. Careltta wants to have an annual income of $680, and the amount invested in the corporate bonds must be half that invested in the Treasury bills. Find the amount in each investment.page 118. The following diagram shows traffic flow in a section of the city. The arrows indicate one-way streets. The numbers on thediagram show how many cars enter or leave this section of thecity via the indicated street in a certain one hour period. Thevariables and represent the number of cars thatBß Cß