Name
Math 311
Exam 1
Spring 2010
Section 502
1.
25 points
P. Yasskin
1
/25
4
/10
2
/25
5
/20
3
/25 Total
/105
Consider the matrices
1 0 1 1
A=
0 1 1 0
1 0 1 0
0 2 3 3
a p
X=
b q
c r
d s
1
and
B=
1
1 −1
1 −1
1
1
a. Compute A −1 .
b. Solve the equation
AX = B.
1
2.
25 points
Consider the system of equations:
3x + 5y − 2z = 17
x+ y
=5
3y − 3z = p
a. Write out the augmented matrix and row reduce it to reduced row eschelon form. (Give reasons
for each step.)
b. For what value(s) of p are there
no solutions?
exactly two solutions?
(At least one answer is "No p".)
a unique solution?
infinitely-many solutions?
c. For those p’s for which there are solutions, what are the solutions?
2
3.
25 points
Consider the system of equations:
3x + 5y − 2z = 17
x+ y
=5
2y − z = p
a. Write out the augmented matrix and row reduce it to reduced row eschelon form. (Give reasons
for each step.)
b. For what value(s) of p are there
no solutions?
exactly two solutions?
(At least one answer is "No p".)
a unique solution?
infinitely-many solutions?
c. For those p’s for which there are solutions, what are the solutions?
3
4.
10 points
Write out the equations which need to be solved to find a, b, c and d to balance the
chemical equation:
a Ca O 2 H 2 + b H 3 P O 4  c Ca 3 P 2 O 8 + d H 2 O
Then set up the augmented matrix for the equations. Do not solve the equations.
5.
20 points
Say whether each of the following statements is true or false for all scalars a, b and c,
all 3 × 3 matrices A, B and C and all 3 × 3 elementary matrices E 1 , E 2 and E 3 of types I, II and III
respectively where E 1 , E 2 and E 3 are NOT the unit matrix. Circle your answers.
a.
b.
c.
d.
e.
f.
g.
h.
i.
j.
A + BC = AC + BC............................................................ True
A + B 2 = A 2 + 2AB + B 2 ....................................................True
AB 2 = A 2 B 2 ....................................................................... True
If B = A − A T then B T = −B................................................ True
AB T = A T B T ...................................................................... True
AB T = B T A T ...................................................................... True
AB −1 = A −1 B −1 ................................................................... True
AB −1 = B −1 A −1 ................................................................... True
aA + bB T = aA T + bB T .................................................True
aA + bB −1 = aA −1 + bB −1 ............................................... True
T
k. E 1 A = AE 1 ............................................................................ True
l. E 2 A =
m. E 3 A =
A T E T2 .......................................................................... True
T
A T E T3 ...................................................................True
n. detcA = c 3 det A................................................................. True
o. detAB −1  = det A − det B......................................................True
False
False
False
False
False
False
False
False
False
False
False
False
False
p. det AB T
False
False
False
q.
False
r.
s.
t.
= det A det B........................................................ True
detE 1 A = 1 .................................................................True
det A
detE 2 A = k det A with k ≠ 1...............................................True
detE 3 A = k det A with k ≠ 1...............................................True
−1
detE 1 E 2 E 3 AE −1
3 E 2  = − det A.............................................True
False
False
False
4