4.3 Row operations. As we have seen in Section 4.1 we can simplify a system of equations by either:

Transcription

1 4.3 Row operations As we have seen in Section 4.1 we can simplify a system of equations by either: 1. Swapping the order of the equations around. For example: can become 3x 1 + 7x 2 = 9 x 1 2x 1 = 2 x 1 2x 1 = 2 3x 1 + 7x 2 = 9 without changing the solution. 2. Multiplying one or more of the equation by a constant. For example: can become x 1 + 2x 2 = x 1 2x 1 = 2 116

2 if we multiply the second equation by Finally, and most importantly, we can subtract or add a multiple of one equation, from another. For example x 1 2x 1 = 2 3x 1 + 7x 2 = 9 can become if we subtract 3 times the first equation, from the second. In all the above cases, we obtained an equivalent system of equations, in the sense that it had the same solution(s), however, it was a simpler system to solve. Note that going from one system, to an equivalent one, changes the augmented matrix form of the system. We can view the change in the corresponding augmented matrix as a row operation that was performed on the augmented matrix. 117

3 Example 4.6. Foreachofthe3examplesabove,write down the corresponding augmented matrices and the row operations needed to go from one to the other. 118

4 4.4 Solving linear equations We have just seen how row operations can be used to simplify a system of equations. How do we know which row operations to perform? How do we know when our system is simple enough? We will learn this now. First, we need a few definitions: Definition 4.7.Let A be a matrix. A leading entry is the first (from the left) non-zero term of each row of A. A leading row (respectively leading column) is a row (respectively column) that contains a leading term. Example 4.8. Indicate the leading entries, rows and columns of the following matrices or augmented matrices. 119

5 Definition 4.9. A matrix is in row-echelon form if all leading rows are above the non-leading rows, and every leading term is to the right of all the leading terms above it. Example Are the following matrices in rowechelon form? 120

6 Solvingasystemoflinearequationsisatwostepprocess. Step 1: Using row operations (also known as Gaussian elimination), put the corresponding augmented matrix in row-echelon form. Step 2: Once in row-echelon form, solve the corresponding system, starting from the bottom most equation. This step is called back substituting Gaussian elimination The following steps describe how to perform Gaussian elimination. 1. Find the first leading column. 2. Swap rows such that the smallest(in absolute value) non-zero number is the highest leading entry in the column. 3. Use this smallest leading entry, to eliminate all the leading entries below it. 4. Go the next leading column, and repeat, until you have traversed all the columns. 121

7 It is important that you always write down your row operations! Example Solve the system of linear equations from page

8 Example Solve the following system of linear equations. x 1 + 2x 2 + 2x 3 + x 4 = 1 2x 1 3x 2 x 3 4x 4 = 0 2x 1 + 5x 2 + 8x 3 + 3x 4 = 3 123

9 Example Solve the following system of linear equations. x 1 + 6x 2 + 5x 3 = 4 x 1 + 7x 2 + 7x 3 = 7 x 1 + 6x 2 + 8x 3 = 6 2x 1 12x 2 10x 3 = 7 124

10 Beforewefinishoffthissection,wehavetolearnonemore concept. We will not really need to use it much in this chapter, but it will be important later on. Definition 4.14.A matrix is in reduced row echelon form if 1. it is in row-echelon form, 2. every leading entry is 1, 3. every leading entry is the only non-zero entry in its column. If an augmented matrix is in reduced row-echelon then to solve the corresponding system of linear equation, back substituting is not required; you can just read off the answer from the matrix. 125

11 Example Put the matrix from Example 4.11 in reduced row-echelon form and solve the corresponding system. 126

12 4.5 Number of solutions As we have seen, every time we solve a linear system of equations we either obtain, no solutions, exactly one solution (unique solution) or infinitely many solutions. You can tell, how many solutions the system has, through the following method by following these steps: 1. Rewrite your system in augmented matrix form. 2. Row reduce, to obtain a row-echelon form (U y). 3. If y is a leading column, then the system has no solutions. If y is not a leading column then If every column of U is leading then there is only one solution. IfU hasnon-leadingcolumns,thenthesystem has infinitely many solutions. IMPORTANT: Having(or not having) rows which are all zeros does not affect the number of solutions!! 127

13 Example Determine the number of solutions to the systems corresponding to the following:

14 Example Findconditionsonb 1,b 2,b 3,b 4 forthe following system to be consistent (i.e. to have solutions). x 1 + 2x 2 + 3x 3 = b 1 2x 1 3x 2 x 3 = b 2 x 1 + 4x x 3 = b 3 4x x x 3 = b 4 129

15 Example Forwhichvaluesofλdotheequations x 1 + 3x 3 = 5 x 1 + λx 2 2x 3 = λ 5 2x 1 λx 2 + (λ+4)x 3 = 9 have a) a unique solution, b) no solution, c) infinitely many solutions? 130