MA122 Mock Final Exam
Name:
**** Please remember that mock tests are meant as a means of providing an extra set of practice questions
and basis for a review class. Do not study for the exam based solely on the topics covered by the mock
test! Go back through notes/assignments/homework to ensure you have reviewed all concepts discussed in
the course.
Time Allowed: 150 minutes
Total Value: 95 marks
Number of Pages: 8
Instructions:
Cheat Sheet:
One 8:5" 11" page of study notes (both sides) is allowed as a reference
while completing the mock test. Please note, that the cheat sheet is permitted
for the mock test only!!
Use of the Casio FX-300MS Plus Calculator is permitted. No other aids allowed.
Answer in the spaces provided.
Show all your work. Insu¢ cient justi…cation will result in a loss of marks.
1. [8 marks] Consider the matrix:
(a) Determine A
1
2
1
A=4 1
0
1
1
5
3
2
9 5.
1
.
2 3
1
!
! 4 5
!
(b) Use your answer to part (a) to determine the solution to A x = b if b = 2 .
3
1
2. [12 marks]
(a) Evaluate the determinant of matrix A where:
2
1
6
6 0
6
A=6
6 2
4
3
3
1
2
4
6
7
2
3
0
3
7
17
7
7
3 7
5
8
(b) Use your answer to part(a) to determine the value of det (2A), where A is the same 4
matrix given in part(a).
82 3 2 3 2
1
3
>
>
<6 7 6 7 6
0
6 7;6 2 7;6
(c) Is the set
4 25 4 65 4
>
>
:
3
7
3 2
1
6
47
7;6
25 4
3
39
0 >
>
=
17
7
linearly independent?
3 5>
>
;
8
Justify your answer, using the results of part(a).
2
4
3. [7 marks] Let M be an invertible n
n matrix.
(a) Show that det M M T > 0.
(b) Show that det M
1
=
1
.
det (M )
2
3
6
4. [5 marks] Given matrix A = 40
0
0
4
1
3
0
7
15, express A as a product of elementary matrices.
0
3
5. [4 marks] Consider the polynomial p (x) = x2
Suppose that B is an n
Show that B
1
3x + 2.
n matrix satisfying the polynomial equation; i.e., p (B) = 0.
exists, and …nd an expression for B
1
(in terms of B and In ).
6. [8 marks] Let L be a linear mapping with the following matrix representation, which can be reduced
to row-echelon form as shown:
3
2
2
2
3 1 0 0
3
2
4 3
1 6
57
7
0 1 0
2 1 0 5~6
[L] = 41
65
4
0
6 1 4
0 0 1
1
(a) State the domain and codomain for the linear mapping L.
Domain:
Codomain:
(b) Give a basis for the range of L.
(c) Give a basis for the rowspace of [L].
(d) Determine the nullity([L]), justifying your answer.
4
7. [14 marks] Consider the two mappings, T : R3 ! R3 and S : R3 ! R3 de…ned by:
T : orthogonal projection on the yz-plane,
S (x1 ; x2 ; x3 ) = (x3 ; x1 + x2 + x3 ;
x2 ).
(a) Show that S is a linear mapping.
(b) State the standard matrix respresentation for the composition S
(c) Prove that S is invertible.
(d) Find the inverse mapping of S, S
1
(w1 ; w2 ; w3 ).
5
T.
8. [4 marks] Given the following matrices,
A=
"
3
0
1
1
determine, if possible, (A
2
4
#
B=
"
1
0
1
0
1
1
#
C=
"
2
1
0
1
1
0
#
,
T
2B) C.
9. [4 marks] A square matrix A is called skew-symmetric if AT =
A.
Prove that if B and C are skew-symmetric matrices, then the matrix kB C is also skew-symmetric
for any scalar k.
2
3
2 3
2
1
10. [4 marks] Determine the volume of the parallelepiped formed by the vectors !
u = 4 15, !
v = 4 35
0
1
2 3
0
amd !
w = 455.
3
6
11. [5 mark] Let unit vectors ~u and ~v in R2 be orthogonal. Suppose that ~x = a~u + b~v for some
a; b 2 R. Prove that ~x ~u = a and ~x ~v = b.
12. [5 marks] Suppose that !
v is an eigenvector for both matrix A and matrix B, with corresponding
eigenvalue
for A and
for B. Show that !
v is also an eigenvector for the matrix AB and
…nd its corresponding eigenvalue.
2
3
13. [15 marks] Consider the matrix A = 4 2
0
2
3
0
(a) Determine all eigenvalues of A.
7
3
0
05.
5
#13, continued.
2
3
A=4 2
0
2
3
0
3
0
05
5
(b) Find an invertible matrix P and a diagonal matrix D such that A = P DP
(c) Use your answers to part(b) to calculate A4 .
8
1
.