Name (Print)
THE UNIVERSITY OF MANITOBA
DEPARTMENT OF MATHEMATICS
I understand that cheating is a serious offense.
136.130 Vector Geometry and
Linear Algebra
Mid-Term Exam
Signature:
Date: Friday, October 26, 2001
Time: 5:306:30 PM
Student Number
DO NOT W
THE UNIVERSITY OF MANITOBA
DATE: October 25, 2004
Midterm Examination
DEPARTMENT & COURSE NO. 136.130
PAGE NO: 1 of 6
EXAMINATION: Vector Geometry & Linear Algebra
TIME: 1 Hour
Midterm exam, 136.130, October 2004: Brief Solutions
(10)
1. Solve, by Gauss-J
1
136.130: Test #5
Solutions
B13.
[7]
1. Which of the following is a subspace of the vector space M 2,2 of all 2 2 matrices (with
the usual addition and scalar multiplication)? Do NOT justify your answer.
a
(a) All matrices of type
a
1
(b) All matrices o
1
136.130: Test #3
Solutions
B15.
1. Solve the following system using Cramers rule. No points will be awarded if other methods are
2x + y = 5
used.
x y = 1
2 1
5 1
2 5
det(A1 ) 6
Solution. With A =
, A1 =
and A2 =
, we have x = det(A) = 3 = 2
1 1
THE UNIVERSITY OF MANITOBA
DATE: October 28, 2002
MIDTERM EXAMINATION
PAGE 1 of 6
DEPARTMENT & COURSE NO: 136.130
TIME: 1 hour
EXAMINATION: Vector Geometry & Linear Algebra
EXAMINERS: Various
_
Values
[10]
1.
Consider the following system of equations:
2
136.130: Test #1 Solutions
B01.
1. Which of the following matrices are in Reduced Row-Echelon Form, which are in RowEchelon Form, and which are neither in RREF nor in REF? (No need to justify your answers for
this question.)
,
,
,
.
Solution. A is in REF
2. 136.130: Test #2
20 minutes
Name:_ Student Number: _
1. Suppose
and
. Find
(a)
(b)
2. Which of the following matrices are in Row Reduced Echelon form, which are
in Echelon form (but not in RRE form) and which are neither? Brieffly
justify your answers.
1
136.130: Test #4 Solutions
4.
1. Find the rank of the following matrix.
- 1 0
(a) - 1 0 - 1 0 -
2
2
2
- 1 0 - 2
This matrix can obviously be row reduced to 0 0 0 so that the row reduced
0 0 0
echelon form will have one non-zero row. So the rank of t
136.130: Test #2 Solutions
B02.
1. Find the inverse of each of the following two matrices, or show it does not exist.
,
.
Solution.
For A:
A is
after row reducing becomes
; so the inverse of
.
For B:
after row reducing becomes
the right); So, B is not inv
3. 136.130: Test #3
20 minutes
Name:_ Student Number: _
1. Suppose
and
. Compute the following if
possible, or state the reason why the operation could not be performed.
(a)
(b)
(c)
(d)
can not be done since the number of columns in B (4) is not the same
B03.
136.130: Test #3 Solutions
1. Suppose
Solution.
. Compute
and
.
2. Suppose A and B and both 4 by 4 matrices and suppose
Find
and
and
.
.
Solution.
3.
(a) Use cofactors to compute the determinant of the matrix
(Hint: Use a row or a column which makes
136.130: Test #4 Solutions
B04.
1. Suppose
and
. Find
(a)
(b)
Solution.
(a)
(b)
.
2. We are given a point
and a vector
.
(a) Find the point normal equation of the plane through P and orthogonal
to u.
(b) Find the parametric equations of the line through P
1. 136.130: Test #5 Solutions
20 minutes
1. Can Cramers rule be used to solve the system
x only.
Solution. The coefficient matrix is
compute
Then
? If yes, find
. We
. So Cramers rule can be applied. Set
.
.
2. Suppose
Find .
. The adjoint matrix of
is
So
1
136.130: Test #1
Solutions
B13.
1. Use Gauss-Jordan elimination to solve the following system. Show your work
describing your steps. State clearly your final answer.
x +
y z = 0
2y 2z = 2
y + z = 1
Solution. Start with the augmented matrix then row redu