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Math 232, Fall 2007
Midterm 1
Nov. 5, 2007
Last Name:
First Name:
SFU ID:
1. DO NOT LIFT UP THE COVER PAGE UNTIL INSTRUCTED.
2. No calculators are allowed.
3. This test is comprised of 7 pages (including cover page)
4. Once the test begins, please check that all pages are intact.
5. Do ALL questions.
6. Clearly explain your answer. No credit will be given for just writing down the
answer.
7. If the answer space provided is not sufficient, write your answer on the back
of the previous page. Clearly mark the question number.
8. Good luck.
Question Points Score
Total: 50
- (7 points) Compute the determinant of
Show all work.
SOLUTION: We use cofactor expansion along the first row to obtain
Using the formula for the determinant of a 2 ร 2, we get an answer of
So the determinant is 2.
- (10 points) An economy is divided into two sectors with consumption matrix
C =
Find the production level ~x needed to satisfy a final demand
d^ ~ =
[
]
Show all work.
SOLUTION: We must solve the equation
~x = C~x + d.~
Equivalently, we need to solve
(I โ C)~x = d.~
This solution is given by
~x = (I โ C)
d.
Note that
I โ C =
and so by the formula for an inverse of a 2 ร 2 matrix, we have
(I โ C)
โ 1
So
(I โ C)
d =
[
]
- Let
A =
(a) (5 points) Find a basis for Col(A). Show all work.
(b) (6 points) Find a basis for N ul(A).
Solution: To do both parts, we must row reduce A. We first subtract row 1 from row 2 and
then subtract twice row 1 from row 3.
๏ฃซ
Next we subtract 3 times row 2 from row 3:
Notice that columns 1, 2, and 4 are pivot columns. Hence, taking these columns in A, we see
is a basis for Col(A).
Notice that if we wish to find a basis for N ul(A), we must row reduce the augmented matrix.
This is not much different, however, and we obtain:
If we interpret this as the augmented matrix of a system of equations in unknowns x 1 , x 2 , x 3 , x 4 ,
and x 5 , then we see x 3 and x 5 are free. Also, โ 6 x 4 โ 6 x 5 = 0, and so x 4 = โx 5 ; x 2 + x 3 + 2x 4 +
2 x 5 = 0, and so x 2 = โx 3 ; x 1 โ x 2 + x 5 = 0, and so x 1 = โx 3 โ x 5. Hence
N ul(A) =
โx 3 โ x 5
โx 3
x 3
โx 5
x 5
: x 3 , x 5 โ R
In parametric form
N ul(A) = x 3
- (12 points) Let W be the subspace of R
3 with basis
B =
Find the coordinates vector of (^) ๏ฃฎ
5 h โ 15
3 h + 12
with respect to the basis B. Show all work.
Solution: We form the augmented matrix and row reduce:
1 3 5 h โ 15
2 โ 1 3 h + 12
โ 1 2 โ 15
1 3 5 h โ 15
0 โ 7 โ 7 h + 42
0 5 5 h โ 30
1 3 5 h โ 15
0 1 h โ 6
0 0 0
Now we analyze. Let x 1 , x 2 denote the coordinates. Then the second row tells us that x 2 = hโ6.
Since x 1 + 3x 2 = 5h โ 15, we see x 1 = 2h + 3. Thus the coordinate vector is
[
2 h + 3
h โ 6
]
