Vaz_MAT_343_ONLINE_B_Fall_2019.hourogne.Section_1.2_Row_Echelon_Form.pdf - Hassan Ouro-gneni Vaz MAT 343 ONLINE B Fall 2019 Assignment Section 1.2 Row

# Vaz_MAT_343_ONLINE_B_Fall_2019.hourogne.Section_1.2_Row_Echelon_Form.pdf

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Hassan Ouro-gneni Vaz MAT 343 ONLINE B Fall 2019 Assignment Section 1.2 Row Echelon Form due 10/20/2019 at 11:59pm MST 1. (1 point) The reduced row-echelon forms of the augmented matrices of four systems are given below. How many solutions does each system have? 1. 1 0 - 11 0 0 1 0 0 0 0 0 1 0 0 0 0 A. Unique solution B. No solutions C. Infinitely many solutions D. None of the above 2. 0 1 0 - 6 0 0 1 5 A. Infinitely many solutions B. Unique solution C. No solutions D. None of the above 3. 1 0 17 0 1 - 16 A. Infinitely many solutions B. No solutions C. Unique solution D. None of the above 4. 1 0 7 0 0 1 18 0 0 0 0 1 A. Unique solution B. No solutions C. Infinitely many solutions D. None of the above Solution: 1. The 3rd row is equivalent to 0 = 1, thus the system is inconsistent 2.The first column has no pivot, therefore x 1 is a free variable and the system has infinitely many solutions 3. There are no free variables and the system has the unique solution x 1 = 17 , x 2 = - 16 4. The last row is equivalent to 0 = 1, thus the system is incon- sistent Answer(s) submitted: B A C B (correct) Correct Answers: B A C B 2. (1 point) Determine how many pivots each of the following matrices have. 1. 0 1 0 - 6 0 0 1 - 7 A. One Pivot B. Two Pivots C. Three Pivots D. Four Pivots 2. 1 0 8 0 1 3 0 0 0 A. One Pivot B. Two Pivots C. Three Pivots D. Four Pivots 3. 1 0 0 0 - 2 0 1 0 0 8 0 0 1 0 - 5 0 0 0 1 2 A. One Pivot B. Two Pivots C. Three Pivots D. Four Pivots 4. 1 0 0 - 6 0 1 0 9 0 0 1 - 4 A. One Pivot B. Two Pivots C. Three Pivots D. Four Pivots Solution: Recall that the pivots are the first non zero entries in each row. 1. There is one pivot in row 1 (and column 2) and another in row 2 (and column 3), for a total of 2 pivots 2. There is a pivot in row 1 and another pivot in row 2, for a total of 2 pivots. 3. Each row contains a pivot, for a total of 4 pivots 4. Each of the rows contains a pivot, thus there are 3 pivots Answer(s) submitted: B B D C (correct) Correct Answers: B 1

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B D C 3. (1 point) How many free variables does each augmented matrix have? 1. 1 0 0 - 10 - 6 0 1 - 5 0 2 0 0 0 0 0 0 0 0 0 0 A. None B. One C. Two D. Three 2. 1 - 6 9 0 0 0 0 0 0 A. None B. One C. Two D. Three 3. 1 0 0 - 7 9 0 1 0 0 - 2 0 0 1 0 8 A. None B. One C. Two D. Three 4. 1 0 0 2 0 1 0 5 0 0 1 5 A. None B. One C. Two D. Three Solution: Note that to determine pivots and free variables, we only look at the coefficient matrix, i.e. we only look at the columns to the left of the vertical bar. 1. The pivots are in columns 1 and column 2, thus x 3 and x 4 are free variables 2. The only pivot is in column 1, thus x 2 is the only free variable 3. The pivots are in columns 1 through 3, thus x 4 is the only free variable 4. There is a pivot in each column, thus there are no free vari- ables Answer(s) submitted: C B B A (correct) Correct Answers: C B B A 4. (1 point)
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