Elementary Row Operations - Mathematics Home

1 Elementary Row OperationsOur goal is to begin with an arbitrary matrix and apply Operations thatrespect row equivalence until we have a matrix in Reduced Row EchelonForm (RREF). The three Elementary row Operations are: (Row Swap) Exchange any two rows. (Scalar Multiplication) Multiply any row by a constant. (Row Sum) Add a multiple of one row to another do these preserve the linear system in question? Swapping rows isjust changing the order of the equations begin considered, which certainlyshould not alter the solutions. Scalar multiplication is just multiplying theequation by the same number on both sides, which does not change thesolution(s) of the equation. Likewise, if two equations share a commonsolution, adding one to the other preserves the is a very simple process for row reducing a matrix, working columnby column. This system is calledGauss-Jordan If all entries in a given column are zero, then the associated variable isundetermined.

2 Make a note of the undetermined variable(s) and thenignore all such Swap rows so that the first entry in the first column is Multiply the first row by so that the pivot is Add multiples of the first row to each other row so that the first entryof every other row is Now ignore the first row and first column and repeat steps 1-5 untilthe matrix is in 9x1+5x2 2x3= 213x1+2x2= 3 First we write the system as an augmented matrix:1 0 0391 5 2213203 R1 R2 132031 5 220 039 3R1 1 6091 5 220 039 R2=R2 R1 16090 1 2 70039 R2 1 6 090 1 270 0 39 R1=R1 6R2 1 0 12 330 1270 039 13R3 1 0 12 330 1270 013 R1=R1+12R3 1 0 030 1 270 0 13 R2=R2 2R3 1 0 030 1 010 0 13 Now we re in RREF and can see that the solution to the system is givenbyx1= 1,x2= 3, andx3= 1; it happens to be a unique solution. Noticethat we kept track of the steps we were taking; this is important for checkingwork!

3 2 Example 10 12 1111 120 1 23 352 141 R2 R1;R4 5R2 10 12 1012 330 1 23 3024 66 R3+R2;R4 2R3 1 0 12 10 12 330 00000 0000 Here the variablesx3andx4are undetermined; the solution is notunique. Setx3= andx4= . Then we can writex1andx2in terms of and as follows:x1= 2 1x2= 2 + 3 + 3We can write the solution set with vectors like so: x1x2x3x4 = 1300 + 1 210 + 2301 This is our preferred form for writing the set of solutions for a linearsystem with many of Gauss-Jordan EliminationTheorem Elimination produces a unique augmentedmatrix in Alice and Bob compute the RREF for a linear system but getdifferent results,AandB. Working from the left, discard all columns exceptfor the pivots and the first column in whichAandBdiffer. By the exercise,3removing columns does not affect row equivalence. Call the new, smaller,matrices Aand B. The new matrices should look this: A=(IdNa00)and B=(IdNb00), whereIdNis anNxNidentity matrix if Aand Bhave the same solution, then we must havea=b.

4 Butthis is a contradiction! ThenA= , Chapter One, Section and , Systems of Linear EquationsReview Problems1. Explain why row equivalence is not affected by removing columns. Isrow equivalence affected by removing rows? Prove or give a (Gaussian Elimination) Another method for solving linear systems isto use row Operations to bring the augmented matrix to row-echelonform. In row echelon form, the pivots are not necessarily set to one,and we only require that all entries left of the pivots are zero, notnecessarily entries above a pivot. Provide a counterexample to showthat row-echelon form is not a system is in row echelon form, it can be solved by back sub-stitution. Write the following row echelon matrix as a system ofequations, then solve the system using back-substitution. 2 3 161 1233 3. Explain why the linear system has no solutions: 1311 246 4 For which values ofkdoes the system below have a solution?

5 X 3y= 6x+3z= 32x+ky+(3 k)z= 15