First and last name:
Open book (but don't use the method from the book), calculators allowed.
1. [10 points ] Solve the system of equations
+4 = 13
4 ,2 + = 7
2 ,2 ,7 = ,19
by writing the corresponding augmented matrix and using elementary row
operations. I want to see you write the steps that you perform, e.g., 2 , 3 1,
to illustrate that you've understood the method. Please present your solution
clearly, preferably in the same way that I use in lectures.
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The augmented matrix is (2 points)
0 1 0 4 13 1
@ 4 ,2 1 7 A
2 ,2 ,7 ,19
There is already a 1 in the top left position, so the row operations leading to zeros
in the rst column are simply
0 1 0 4 13 1
1 0 4 13
1 7 to get @ 0 ,2 ,15 ,45 A
2 , 4 1 4 ,2
2 ,2 ,7 ,19
2 ,2 ,7 ,19
(1 point for correct operation, 1 point for correct arithmetic), then
0 1 0 4 13 1
1 0 4 13
0 ,2 ,15 ,45 to get @ 0 ,2 ,15 ,45 A
,7 ,19
0 ,2 ,15 ,45
3 , 2 1 2 ,2
(1 point for correct operation, 1 point for correct arithmetic). Finally, to get a zero
in the third row/second column, do
0 1 0 4 13 1
1 0 4 13
0 ,2 ,15 ,45 to get @ 0 ,2 ,15 ,45 A
,7 ,19
0 0 0
0
3 , 2 2 ,2
(1 point for correct operation, none for arithmetic!). Now it's in echelon form,
rewrite this as a system of equations (1 point):
+4 = 13
,2 ,15 = ,45
Use back substitution: solve for and in terms of to get = , 452 + 152 and
= 13 , 4 (1 point). Hence the answer is
15 ) : 2 Rg
(
) = f(13 , 4 , 45
+
2 2
(2 points; you lose these two points if your solution does not convey the fact that
may take any real value).
Solution:
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