Math 115 Midterm Review Problems
1.
(a) Solve the following linear system:
x
+
y
+
2
z
=
2
2
x
+
y

z
=
3
x
+
2
y
+
7
z
=
3
(b) Find the solution to the associated homogeneous system:
x
+
y
+
2
z
=
0
2
x
+
y

z
=
0
x
+
2
y
+
7
z
=
0
2. Suppose
A
,
B
and
C
are invertible
n
×
n
matrices. Prove the following, using the definition of matrix
inverse:
(
A

1
BC
)

1
=
C

1
B

1
A
3. Find the values of the number
c
such that
1
c
0
2
0
c
c

1
1
has an inverse.
4. Find the inverse of
1

1

2

1
0
1
2
1
0
,
and use it to solve the system
x

y

2
z
=
3

x
+
z
=
0
2
x
+
y
=
1
5. Consider the transformation
T
:
R
2
→
R
2
defined as follows:
Counterclockwise rotation about the origin through an angle of
π/
2 followed by reflection in the line
y
=
x.
Determine the standard matrix for
T
and find
T
1
2
.
6. If
T
A
:
R
2
→
R
2
is a linear transformation defined by
T
A
x
y
=
x
+
y
x

y
and
T
B
:
R
2
→
R
2
is a linear
transformation defined by
T
B
x
y
=
3
x
2
x
+ 4
y
, find
(a) the standard matrix for
T
A
◦
T
B
.
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 Fall '07
 DUNBAR
 Math, Linear Algebra, Algebra, TA, standard matrix

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