ࡱ>  Z{[ -dbjbj ůΐΐ[88F F !!!$EEEPE~mI/_/_//@@@$:>!@=Y@^@@_//:III@l_/(!/I@IIV@,Q/G#EiA0Y xCGlxQQx!l@@I@@@@@H@@@@@@@x@@@@@@@@@F O: Mat 211 Dr. Firoz 2-3: System of equations and matrices  Systems, Matrices, and Applications Systems of Linear Equations System of equation  (Has solution) Consistent Inconsistent (has no solution)  Dependent Independent For Example: Consider the system  EMBED Equation.DSMT4  Solve it and see that it has a unique solution. The system is consistent and independent. The system  EMBED Equation.DSMT4  Is inconsistent, it has no solution. Check the solution. Now the system  EMBED Equation.DSMT4  has infinitely many solutions, it is consistent and dependent. We can solve it by the following methods: Algebraically Using Elimination Algebraically Using Substitution Using Graphical method Augmented Matrices and Row Operations. Matrix algebra (the  EMBED Equation.DSMT4  form) Cramers Rule (Determinants). Definition: The following representation of two linear equations is called the system of linear equations in two variables:  EMBED Equation.3   EMBED Equation.3  Solution: A solution of the given system is an ordered set or list of numbers that satisfies all the equations simultaneously when we put  EMBED Equation.3 and EMBED Equation.3 . This solution is written as  EMBED Equation.3 . We have our solution in this case as  EMBED Equation.3 . If the given system has at least one solution, it is said to be consistent. When the system has no solution, it is said to be inconsistent. Examples The system  EMBED Equation.3  ,  EMBED Equation.3  has unique solution. Find the solution. The system  EMBED Equation.3 ,  EMBED Equation.3  has infinitely many solutions.. Write the solution in terms of the third variable. The system  EMBED Equation.3 ,  EMBED Equation.3  is inconsistent. Solve the system  EMBED Equation.3 ,  EMBED Equation.3  If the system  EMBED Equation.3 ,  EMBED Equation.3  is inconsistent find k. Solve the system  EMBED Equation.3 ,  EMBED Equation.3 ,  EMBED Equation.3  Consider this 2 by 2 system: use all the methods discussed above to solve the system:  EMBED Equation.DSMT4  Solution: Algebraically Using Elimination  EMBED Equation.DSMT4  EMBED Equation.DSMT4  EMBED Equation.DSMT4  By substituting  EMBED Equation.DSMT4 we get  EMBED Equation.DSMT4  Algebraically Using Substitution  EMBED Equation.DSMT4  Using Graphical method  EMBED Equation.DSMT4  Augmented Matrices and Row Operations.  EMBED Equation.DSMT4  The last row gives  EMBED Equation.DSMT4 , and by backward substitute we get  EMBED Equation.DSMT4  Answers: 1. (1, 1) 2. ( EMBED Equation.3 ) 3. No solution 4. (-2, 6) 5. k = 6 6. (1, 1,1) Example 2. Lets see another example by using Augmented Matrices and Row Operations. Solve for x, y and z  EMBED Equation.DSMT4  or  EMBED Equation.DSMT4  Augmented Matrix:  EMBED Equation.DSMT4  Multiply row 1 by 1 and add with row 2 and get your new row 2  EMBED Equation.DSMT4  ;  EMBED Equation.DSMT4  Multiply 2 with row 1 and add with row 3 and get your new row 3  EMBED Equation.DSMT4 ;  EMBED Equation.DSMT4  Multiply row 2 by 1/3 gives you following  EMBED Equation.DSMT4  ;  EMBED Equation.DSMT4  Multiply by -2 with row 2 and add with row 3 and get your row 3  EMBED Equation.DSMT4 ;  EMBED Equation.DSMT4  Now by backward substitution to solve for variables:  EMBED Equation.DSMT4   EMBED Equation.DSMT4  Now practice:  EMBED Equation.DSMT4  ( Write the system of equations and solve. Ans (2, -1, 1) Reduced Row Echelon Form (rref): The Elimination Method for solving large systems of linear equations Make the leading coefficient 1 either by interchanging row or by multiplying or dividing the first by a suitable constant. Eliminate the leading coefficient each later equation by replacing the later equation by the sum itself and a suitable multiple of the first equation Repeat step 1 and 2for the 2nd row to eliminate leading coefficient 0 and to make 2nd element 1 Continue the process until all diagonal elements are 1 then do back substitution to solve variables. Example 3. Below are three row-reduced echelon forms for matrices of certain linear systems. For each matrix, tell how many solutions the system has. Explain. If there are any, find the solutions. If there are infinitely many solutions, find the general formula and two particular solutions. a.  EMBED opendocument.MathDocument.1 The system has unique solution with x=3, y=-2, z=5 b.  EMBED opendocument.MathDocument.1  The system does not have solution c.  EMBED opendocument.MathDocument.1  The system has many solutions, it is dependent and consistent. The general solutions are  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  Now lets discuss system of equations again: Matrix algebra (the  EMBED Equation.DSMT4  form) Example: let  EMBED Equation.DSMT4  For matrix algebra we also use this inverse concept to solve systems of equation. Singular Matrix: Singular matrices do not have an inverse. A matrix is singular matrix if determinant of the matrix is equal to zero, let A is a matrix then  EMBED Equation.DSMT4 exist if  EMBED Equation.DSMT4 . So lets discuss how to find determinant of a matrix Finding determinant If  EMBED Equation.DSMT4  Example:  EMBED Equation.DSMT4  Therefore A is not singular matrix. It has inverse. Find the inverse. And verify that  EMBED Equation.DSMT4 . What is  EMBED Equation.DSMT4 ? Answer: I is an identity matrix. Finding Determinant of a 3 by 3 matrix  EMBED Equation.DSMT4   Note: The determinant of a matrix will be zero if An entire row is zero. Two rows or columns are equal. A row or column is a constant multiple of another row or column. Remember, that a matrix is invertible, non-singular, if and only if the determinant is not zero. So, if the determinant is zero, the matrix is singular and does not have an inverse. Sarrus Rule. In order to compute  INCLUDEPICTURE "http://kr.cs.ait.ac.th/%7Eradok/math/mat3/m112.jpg" \* MERGEFORMATINET   INCLUDEPICTURE "http://kr.cs.ait.ac.th/%7Eradok/math/mat3/m113.jpg" \* MERGEFORMATINET  Note: Sarrus rule is only applicable if the determinant is of order 3 by 3. Example 1. Use Sarrus rule to find the value of  EMBED Equation.DSMT4   12 + 4 + 48 = 64  EMBED Equation.DSMT4  24 + 3 + 32 = 59 Exercise 1. Use Sarrus rule to evaluate the determinant  EMBED Equation.DSMT4  and verify your answer using calculator. Finding inverse of a matrix: Now lets learn how to find inverse of a matrix. There are different methods to find inverse matrix. Method 1. Use Shortcut for 2 by 2 matrix Let  EMBED Equation.DSMT4 then  EMBED Equation.DSMT4  Example:  EMBED Equation.DSMT4 then  EMBED Equation.DSMT4  Method 2 (Optional). Use  HYPERLINK "http://www.mathwords.com/g/gauss-jordan_elimination.htm" Gauss-Jordan elimination to transform [ A | I ] into [ I | A-1 ]. Example: Consider a matrix  EMBED Equation.DSMT4  and write the following (use rref using calculator)  INCLUDEPICTURE "http://www.mathwords.com/i/i_assets/inverse%20of%20a%20matrix%20example%203b.gif" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.mathwords.com/i/i_assets/inverse%20of%20a%20matrix%20example%203c.gif" \* MERGEFORMATINET  Method 3 (Optional).  HYPERLINK "http://www.mathwords.com/a/adjoint.htm" Adjoint method A-1 =  INCLUDEPICTURE "http://www.mathwords.com/i/i_assets/inverse%20of%20a%20matrix%201%20over%20det%20A.gif" \* MERGEFORMATINET (adjoint of A) or A-1 =  INCLUDEPICTURE "http://www.mathwords.com/i/i_assets/inverse%20of%20a%20matrix%201%20over%20det%20A.gif" \* MERGEFORMATINET ( HYPERLINK "http://www.mathwords.com/c/cofactor_matrix.htm" cofactor matrix of A)T Let  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  Now we know how to find inverse, lets go back to solution of system of equations: Example 1. (continued) Given system is  EMBED Equation.DSMT4  If we write in matrix form then we get the following,  EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4  If we do  EMBED Equation.DSMT4 , we get the given system and we can rewrite  EMBED Equation.DSMT4  We have seen that  EMBED Equation.DSMT4 then  EMBED Equation.DSMT4  Now  EMBED Equation.DSMT4  so  EMBED Equation.DSMT4 and  EMBED Equation.DSMT4  Cramers Rule (Determinants) Given the system  INCLUDEPICTURE "http://www.okc.cc.ok.us/maustin/Cramers_Rule/Image994.gif" \* MERGEFORMATINET  This system has the unique solution  INCLUDEPICTURE "http://www.okc.cc.ok.us/maustin/Cramers_Rule/Image995.gif" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.okc.cc.ok.us/maustin/Cramers_Rule/Image996.gif" \* MERGEFORMATINET  Example 1. (Continued) Solve (using Cramers Rule)  EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4  Now,  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  Example 2. Use Cramers Rule to solve the system: 4x - y + z = -5 2x + 2y + 3z = 10 5x 2y + 6z = 1 Solution. We begin by setting up four determinants:  INCLUDEPICTURE "http://www.okc.cc.ok.us/maustin/Cramers_Rule/Image1013.gif" \* MERGEFORMATINET : D consists of the coefficients of x, y, and z from the three equations  INCLUDEPICTURE "http://www.okc.cc.ok.us/maustin/Cramers_Rule/Image1014.gif" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.okc.cc.ok.us/maustin/Cramers_Rule/Image1016.gif" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.okc.cc.ok.us/maustin/Cramers_Rule/Image1018.gif" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.okc.cc.ok.us/maustin/Cramers_Rule/Image1020.gif" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.okc.cc.ok.us/maustin/Cramers_Rule/Image1015.gif" \* MERGEFORMATINET is obtained by replacing the x-coefficients in the first column of D with the constants from the right sides of the equations.  INCLUDEPICTURE "http://www.okc.cc.ok.us/maustin/Cramers_Rule/Image1017.gif" \* MERGEFORMATINET is obtained by replacing the y-coefficients in the second column of D with the constants from the right sides of the equations.  INCLUDEPICTURE "http://www.okc.cc.ok.us/maustin/Cramers_Rule/Image1019.gif" \* MERGEFORMATINET is obtained by replacing the z-coefficients in the third column of D with the constants from the right sides of the equations. Next, we evaluate the four determinants:  INCLUDEPICTURE "http://www.okc.cc.ok.us/maustin/Cramers_Rule/Image1021.gif" \* MERGEFORMATINET  = 4(12 (-6)) + 1(12 15) + 1(-4 10) = 4(18) + 1(-3) + 1(-14) = 72 3 14 = 55  INCLUDEPICTURE "http://www.okc.cc.ok.us/maustin/Cramers_Rule/Image1022.gif" \* MERGEFORMATINET  = -5(12 (-6)) + 1(60 3) + 1(-20 2) = -5(18)+1(57) + 1(-22) = -90 + 57 22 = -55  INCLUDEPICTURE "http://www.okc.cc.ok.us/maustin/Cramers_Rule/Image1023.gif" \* MERGEFORMATINET  = 4(60 3) + 5(12 15) + 1(2 50) = 4(57) + 5(-3) + 1(-48) = 228 - 15 48 = 165  INCLUDEPICTURE "http://www.okc.cc.ok.us/maustin/Cramers_Rule/Image1024.gif" \* MERGEFORMATINET  = 4(2 (-20)) + 1(2 50) 5(-4 10) = 4(22) + 1(-48) 5(-14) = 88 48 + 70 = 110 Substitute these four values into the formula from Cramers Rule:  INCLUDEPICTURE "http://www.okc.cc.ok.us/maustin/Cramers_Rule/Image1025.gif" \* MERGEFORMATINET  So, the solution is (-1, 3, 2). Matrix Algebra on the Calculator TI: 1. Hit 2nd-MATRIX, then EDIT, and select [A], [B], etc. 2. Type in the size(order) of the matrix and the entries. Remember to hit ENTER after each entry. 3.Repeat these steps until you have all your matrices entered. 4. To calculate hit 2nd-QUIT, then hit 2nd-MATRIX, then simply select [A]. Then type the x-1 key, then hit ENTER. To convert this to fractions, hit MATH, then select Frac, and hit ENTER. 5. To calculate A-1B, do as in step 4, but select [B] also, so that you have [A]-1[B] on your screen. Hit ENTER to get the answer. CASIO: 1. Hit MENU, and select 3:MAT. Select the matrix, type in its size and the entries until you have all your matrices finished. 2. To calculate A-1, hit MENU, and select 1:RUN. Hit OPTN, then select F2:MAT, then F1:MAT. This will bring up a Mat on the screen. Then you hit ALPHA (Red key) and A (just below) to bring up A. Then hit the x-1 key above the ). 3. Repeat these steps for  EMBED Equation.DSMT4 . Section: Gaussian Elimination In this section we will learn a general method for finding possible solutions to a linear system of equations. The method involves systematic elimination of the unknown from each equation in turn. We will explain the method with examples. Example 1. Solve the system  EMBED Equation.3  Solution: Now applying the operation  EMBED Equation.3  we have the following  EMBED Equation.3  Applying  EMBED Equation.3  we have  EMBED Equation.3  And by  EMBED Equation.3   EMBED Equation.3  Finally we the following by applying  EMBED Equation.3   EMBED Equation.3  We now have that  EMBED Equation.3 , and other unknowns can easily be found by backward substitution into second and first equations. We have the solution  EMBED Equation.3 . This method is called the Gaussian Elimination method. Augmented matrix: Let us consider the system of equations  EMBED Equation.3  The augmented matrix of the above system is  EMBED Equation.3  Row Echelon Form: A matrix is in row echelon form when The entry in row 1, column 1 is a 1 and 0 appears below it The first nonzero entry in each row after the first row is a 1, zeros appear below it and it appears to the right of the first nonzero entry in any row above. Any row that contains all zeros to the left of the vertical bar appear at the bottom. The row echelon form of the above augmented matrix is  EMBED Equation.3 which exactly our Gaussian Elimination. Reduced Row Echelon Form (rref): Sometimes it is advantageous to write a matrix in reduced row echelon form. In this form, row operations are used to obtain entries that are zero above as well as below the leading 1 in a row. From reduced row echelon form one can find solution of a system directly without backward substitution. The reduced row echelon form of the above row echelon form will look like the one given below:  EMBED Equation.3 . The left hand side of the vertical bar is the identity matrix. Finding reduced row echelon form by calculator: TI: Hit 2nd  EMBED Equation.3 , then edit your matrix, hit 2nd MODE to go to the normal screen.  Hit 2nd  EMBED Equation.3  then go to MATH, hit 8 for rref (and input your matrix A using again 2nd  EMBED Equation.3 , hit enter. You will get your reduced row echelon form. CASIO: Go to MENU, choose MAT, use right arrow to select size and edit your matrix Go to OPTION and Press F1 for row operation Example 2: Find the value of K so that the given system has no solution:  EMBED Equation.3  Solution: Find the determinant value of the coefficient matrix and set equal to zero.  EMBED Equation.3  Using Sarruss rule see that  EMBED Equation.3 . You may also use row/column operation to find two zeros and find your result from reduced system. Example: Use Cramers Rule to explain why the system  EMBED Equation.3  has no solution.  Section: Matrices and Matrix Operations Matrix: A matrix is simply a rectangular array of numbers considered as an entity. A matrix with m rows and n columns in the array is written as follows:  EMBED Equation.3  and a column vector looks like  EMBED Equation.3 which is of order  EMBED Equation.3  Example 1. Given EMBED Equation.3 ,  EMBED Equation.3  and  EMBED Equation.3 . Find the following results: a)  EMBED Equation.3  b)  EMBED Equation.3  Example 2. Solve for x and y Example 3.  EMBED Equation.DSMT4  Example 4.  EMBED Equation.DSMT4  Order of above matrices: A is a matrix 3 by 2, B is 3 by 3, C is 3 by 2, D is 3 by 1, E is 1 by 3, F and G are 2 by 2 matrices. Find the following if possible -B, 2. 3A-2C, 3. F+3G, 4. 2B-5C Answers:  EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4  2B-5C is not possible  Group work: now try Consider  EMBED Equation.3  Find 2A + 2C, 2A+B Example 6. Given the matrix EMBED Equation.3 . Find the determinant value of  EMBED Equation.3 . Answer is zero. Section: Matrix Multiplication The multiplication of two matrices A and B is a unique matrix AB which is defined for A of order EMBED Equation.3 , while B is of order EMBED Equation.3 . Note that the order of AB is EMBED Equation.3 . Example 1. Given EMBED Equation.3 ,  EMBED Equation.3  and  EMBED Equation.3 . Find the following results: a)  EMBED Equation.3  b)  EMBED Equation.3  c)  EMBED Equation.3  Example 2. Find x  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  Now solve for x (Answer should be 4, -5/4) Example 3. Multiply the matrices  EMBED Equation.3  Example 4. Let  EMBED Equation.3 . Determine the following or explain why the answer does not exist. Product AB. Answer:  EMBED Equation.3  Sum 3A + 2C. Answer:  EMBED Equation.3  Product BC. Answer: Does not exist Sum A + 3B. Answer: Does not exist Matrix Multiplication: Matrix Multiplication is possible if number of column of first matrix is equal to number of row of 2nd matrix. Example :  EMBED Equation.DSMT4  Find the following:  EMBED Equation.DSMT4  EMBED Equation.DSMT4   EMBED Equation.DSMT4  Now try for BC and CB . Properties of addition and scalar multiplication:  EMBED Equation.DSMT4  Exercise 5. From the given matrices find  EMBED Equation.3 , where I is the identity matrix.  EMBED Equation.3  Example 6. Let  EMBED Equation.3 . Calculate  EMBED Equation.3 , then look at the pattern and find  EMBED Equation.3 , for n, a positive integer. Answer:  EMBED Equation.3  Section 7.5 Rules for Matrix Multiplication (More examples Page # 282) Example 1. Suppose P, and Q are  EMBED Equation.3  square matrices such that  EMBED Equation.3 . Prove that  EMBED Equation.3 . Example 2. For the following matrices verify that  EMBED Equation.3 .  EMBED Equation.3  Example 3.  EMBED Equation.3 , show that  EMBED Equation.3 , find also  EMBED Equation.3  Example 4. Prove that if  EMBED Equation.3 , then  EMBED Equation.3 , where I is a 2 by 2 identity matrix. Section: The Inverse of a Matrix For a given non-singular matrix A, the inverse matrix  EMBED Equation.3  exists such that  EMBED Equation.3 , where I is an identity matrix of order same as A or B. A matrix A is non-singular iff  EMBED Equation.3  To find inverse of a nonsingular matrix using calculator: Step 1. Input the matrix say A Step 2. Call matrix A and hit  EMBED Equation.3  in your calculator then hit MATH and select 1 :  EMBED Equation.3 Frac to get the matrix along with determinant value. Example 1. Find the inverse of a two by two matrix by hand:  EMBED Equation.3  then  EMBED Equation.3  Now verify that  EMBED Equation.3  Example 2. Show that  EMBED Equation.3  Solution: We can consider that  EMBED Equation.3  Example 3. Show that  EMBED Equation.3  using  EMBED Equation.3  Example 4. Solve the following system of equations by matrix inverse:  EMBED Equation.3  Solution: We have the following matrix system  EMBED Equation.3  Example 5. Show that the following system has no solution using inverse matrix method.  EMBED Equation.3  Section: Determinant of order 2 by 2 and Cramers Rule We have discussed about it in the previous lecture. Check your note and also you may go over your test book. Some more exercise problems are given below: Exercise 1. Let  EMBED Equation.3  show that  EMBED Equation.3  Exercise 2. Let  EMBED Equation.3  , calculate  EMBED Equation.3  Exercise 3. Let  EMBED Equation.3  find  EMBED Equation.3  Exercise 4. Show that  EMBED Equation.3  Exercise 5. If  EMBED Equation.3  find all values of a. Answer:  EMBED Equation.3  Exercise 6. If  EMBED Equation.3  determine  EMBED Equation.3  Answer:  EMBED Equation.3  More examples: Determinant: 2 by 2:  EMBED Equation.3  3 by 3:  EMBED Equation.3  Another way of expansion (making two zeros by operation)  EMBED Equation.3  Now verify the following:  EMBED Equation.3  b)  EMBED Equation.3  c)  EMBED Equation.3  Prove the following:  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  Solve for all x, a, b, or c:  EMBED Equation.3  b)  EMBED Equation.3  c)  EMBED Equation.3  Answer: a)  EMBED Equation.3  b)  EMBED Equation.3  c)  EMBED Equation.3  Matrices: Given  EMBED Equation.3  Given  EMBED Equation.3  For a given matrix  EMBED Equation.3 , the transpose is  EMBED Equation.3 , first row becomes first column and second row becomes second column. A matrix A is called orthogonal (perpendicular) if  EMBED Equation.3 . Show that the matrix  EMBED Equation.3  is orthogonal. Given  EMBED Equation.3 . If  EMBED Equation.3 , where O is the zero matrix, find the matrix X. If  EMBED Equation.3 , find x and y Suppose that  EMBED Equation.3  find  EMBED Equation.3  Given  EMBED Equation.3 , find x, y and z Solve the following system using both Cramers rule and matrix inverse:  EMBED Equation.3  b)  EMBED Equation.3  c)  EMBED Equation.3  d)  EMBED Equation.3  e)  EMBED Equation.3  Given an input matrix  EMBED Equation.3  and the demand matrix  EMBED Equation.3 . Find the output matrix X, such that  EMBED Equation.3  Find k so the system has no solution:  EMBED Equation.3  How do you determine whether the system  EMBED Equation.3  has infinitely many solutions or no solution at all? How do you determine whether the system  EMBED Equation.3  has infinitely many solutions or no solution at all? Find an equilibrium vector for the transition matrix given below:  EMBED Equation.3  Solution: The equilibrium vector is a row matrix  EMBED Equation.3  such that  EMBED Equation.3 . Use matrix multiplication and find v.  EMBED Equation.3  The required equilibrium vector is  EMBED Equation.3  which is a 1 by 3 matrix.        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