This preview shows page 1. Sign up to view the full content.
Math 115 2002 Midterm Test
Problem 1.
(a) Determine the values of
a
and
b
for which the following system of linear equations has
(i) exactly one solution,
(ii) inFnitely many solution, and
(iii) no solutions.
x
1
+
ax
3
=2
x
1
+
x
2
+(
a
+1)
x
3
=7
x
1
+
x
2
+2
ax
3
=
b
.
(b) ±or the values of
a
and
b
in part (ii) above, give the general solution to the system.
Problem 2.
(a) A 3
×
3matr
ix
B
has the following elementary row operations performed on it, in the order given:
1)
R
1
→
R
1
+
aR
2
(add
a
t
imesrow2torow1)
2)
R
3
±
R
2
( interchange row 2 and row 3)
3)
R
2
→
bR
2
(multiply row 2 by
b
)
where
a
and
b
are nonzero real numbers.
±ind the matrix
A
such that the matrix product
AB
gives the same result as preforming the
above three elementary row operations on
B
.
(b) ±or the matrices
A
and
B
from part (a) of this question, if det(
B
) = 5, calculate det(
AB
).
Problem 3.
Given that
C
and
D
are 5
×
5 matrices with det(
C
)=
−
2 and det(
D
) = 3. determine the
numerical value of each of the following:
(i) det(
CD
)
(ii) det(
C
t
(iii) det(
D

1
(iv) det(2
D
)
(v) det(
C
3
)
Problem 4.
This is the end of the preview. Sign up
to
access the rest of the document.
This note was uploaded on 05/18/2011 for the course MATH 115 taught by Professor Dunbar during the Spring '07 term at Waterloo.
 Spring '07
 DUNBAR
 Math, Linear Algebra, Algebra, Linear Equations, Equations

Click to edit the document details