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Geogebra classic rref9/24/2023 Gaussian elimination, (a) Using Gaussian elimination, solve the following simultaneous equa- tions by first forming the augmented matrix and bringing it to re- duced rou echelon form. (b) Represent your solutions to the equations in part (a) by using the column picture as introduced in the lectures. You may draw your diagrams in pencil or use a free graphing software such as Geogebra Classic in 3 dimensions. (c) Now consider the system of equations: 2x+y=3 (2) 4x + 2y = 5 1 #GEOGEBRA CLASSIC 5 RREF FREE# Write down the augmented matrix for 2 and show that the re- duced row echelon form is given by: 6:19) This means that Or + Oy = 1(a contradiction). Therefore, the equations have no solution. Display equation 2 in the row picture (i.e. Using the row picture explain why the equations have no solu- tion. Similarly, using the column picture, explain why equation 2 has no solution. You may draw the diagrams by hand or by using drawing software such as Geogebra. (d) By using Gaussian elimination, find the inverse of the coefficient ma- trix in equation 1, part a. Simultaneous equations with infinitely many solutions. (a) Consider the equations: *+ 2y = 1 (3) 2r + 4y = 2 i. Show that the RREF of equation 3 is (630) We therefore have 0x + Oy = 0 = 0 = 0. We can only conclude that there are infinitely many solutions that satisfy r + 2y = 1. Using this, explain why there are many soutions. Using the solutions (1.0) and (-1,1) draw the column picture of 3. Usually, we set one of the variables equal to a real parameter. Write the augmented matrix for these equations and through Gaussian elimination, bring it to reduced row echelon form. Again, this gives us no useful information. Therefore, we have two equations in three variables and ii. In the row picture, each of the equations in equation 4 represents a plane. Explain the relationship between the planes described by the second and third equations in 4. You do not have to draw anything for this part iii.
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