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Unformatted text preview: echelon or staircase-shaped forms. These echelon forms will be studied in the next section. They will also be used for m x n systems, where m-=I- n. EXERCISES 1. Use back substitution to solve each of the following systems of equations. (a) Xl- 3X2- 2 2X2- 6 (b) Xl + X2 + X3- 8 2X2 + X3 5 3X3 9 (c) Xl + 2X2 + 2X3 + X4 5 3X2 + X3- 2X4 1-X3 + 2X4 =-1 4X4- 4 Xl + X2 + X3 + X4 + XS = 5 2X2 + X3 2X4 + XS 4X3 + X4- 2xs X4- 3xs = 2xs- 2 2. Write out the coefficient matrix for each of the systems in Exercise l. 3. In each of the following systems, interpret each equation as a line in the plane. For each system, graph the lines and determine geometrically the number of solutions. (a) Xl + X2 = 4 (b) Xl + 2X2 = 4 2-2XI 4X2 = 4 2Xl- X2 = 3 (d) Xl + X2 6 Xl- X2-Xl + 3X2 = 3 4. Write an augmented matrix for each of the systems in Exercise 3. 5. Write out the system of equations that corresponds to each of the following augmented matrices. (a) [i ~ I ~ ] (b) [ ;-2-~ I ~] 3 1 4-~ ] [!-31 2 i]-2 3 (d) 1-5 6 1 24 6-1 1 3-2 XI + X2 + X3 + 2X4 2XI + 3X2 + X3 + 3X4 X2 + X3 + X4 (b) 2x] + X2 = 8 4x]- 3X2 = 6 (d) XI + 2X2X3 2XI X2 + X3-XI + 2X2 + 3X3 (f) 3xI + 2X2 + X3 o 7 6 6 o 2-1 3 7 4X4-1 X3 + 3X3 X2 + 2X3 2xI + X2 2XI-2xI + X2 3XI and (h) 6. Solve each of the following systems. (a) XI 2X2 = 5 3xI + X2 (c) 4xI + 3X2- 4 ~XI + 4X2 3 (e) 2xI + X2 + 3X3 4XI + 3X2 + 5X3 6xI + 5X2 + 5X3 =-3 (g) I 2 2X3 =-1 3XI + 3X2 + Xl + 2X2 + ~X3- 3 2- 2: I 2 12_ I 2:XI + X2 + SX3- TO e two systems 2xI + X2 = 3 III i 4XI + 3X2 = 5 4xI + 3X2 have the same coefficient matrix but different right-hand sides. Solve both systems simultaneously by eliminating the first entry in the second row of the augmented matrix [2 1 [3-1] 4 3 5 1 and then performing back substitutions for each of the columns corresponding to the right-hand sides. 8. Solve the two systems 2xI + 5X2 + X3 2xI + 5X2 + X3 XI + 2X2 2X3 1 9 XI + 2X2 2X3 9 9 Xl + 3X2 + 4X3 = 9 XI + 3X2 + 4X3 =-2 by doing elimination on a 3 x 5 augmented matrix and then performing two back sub- stitutions. 9. Given a system of the form-mlxI + X2 = bl-m2xI + X2 = b2 where ml, m2, bI, and b2 are constants: (a) Show that the system will have a unique solution if ml =1= m2· (b) If ml = m2, show that the system will be consistent only if h = b2· (c) Give a geometric interpretation to parts (a) and (b)....
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- Winter '11
- Row echelon form, X3 X2 Xl, X3 X2 13Xl8X2+=