The problems listed here arc old quiz/test/exam problems. Print this file, solve
ID: 3037980 • Letter: T
Question
The problems listed here arc old quiz/test/exam problems. Print this file, solve all problems in full detail on your printout, then scan it and upload a pdf of your work on backboard by noon on Friday, April 21 2017. Justify your answers! Consider the linear system: {2x_1 + 5x_2 + 3x_3 + 3x_4 = 3 x_1 + 2x_2 + 2x_4 = 0 - 3x_1 - 5x_2 + 3x_3 - 2x_4 = 3} a Write the augmented matrix for this system. Use elementary row operations to place it in reduced row - echelon form (not just triangular form). Specify the operations used for each step. b. Find all solutions of the system. Express the solution set m vector form. Describe the solution set as a point, line, or plane in R^n stating what n is. c. Based on your work in part (a), what is the lank of the associated coefficient matrix? Explain briefly.Explanation / Answer
(a). The augmented matrix of the given linear system is B =
2
5
3
3
3
1
2
0
2
0
-3
-5
3
-2
3
We can reduce B to its RREF as under:
Multiply the 1st row by ½
Add -1 times the 1st row to the 2nd row
Add 3 times the 1st row to the 3rd row
Multiply the 2nd row by -2
Add -5/2 times the 2nd row to the 3rd row
Multiply the 3rd row by 1/5
Add 1 times the 3rd row to the 2nd row
Add -3/2 times the 3rd row to the 1st row
Add -5/2 times the 2nd row to the 1st row
Then the RREF of B is
1
0
-6
0
-6
0
1
3
0
3
0
0
0
1
0
(b) From the RREF of B, we observe that the given linear system is equivalent to x1 -6x3 = -6 or, x = 6x3-6 , x2 +3x3 = 3 or, x2 = 3-3x3 and x4 = 0. Hence X = (x1,x2,x3,x4)T= (6x3-6, 3-3x3,x3 ,0)T = x3(6,-3,1,0)T +(-6,3,0,0)T. Hence the solution space of the given linear system is span{(6,-3,1,0)T ,(-6,3,0,0)T}. This solution space, being the span of 2 vectors in R3 ,is a plane in R3.
(c ) From the RREF of B, it is apparent that the RREF of the associated coefficient matrix is
1
0
-6
0
0
1
3
0
0
0
0
1
Since there are no zero rows in this matrix, hence the rank of the associated coefficient matrix , being equal to the number of non-zero rows in its RREF, is 3.
2
5
3
3
3
1
2
0
2
0
-3
-5
3
-2
3
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