ASSIGNMENT 5
MATH 270, WINTER 2010.
DUE: 5PM, APRIL 7, 2010.
Assignments must have your name, student number, course number and TA’s name on the first page.
Please submit them by placing them in the box marked “Assignments” outside the Maths Office (Burn
side Hall, 10th floor). Show all working and justifications!
(1) The matrices
A
=
1
4
1

1
1

1

1
1

1
1
1

1
1

1

1
1

1
1
and
B
=
0
0
1
0
0
0
0
1
0
1
0
0
1
0
0
0
belong to which of the following classes:
•
Defective?
•
Factorisable as
A
=
LU
or
B
=
LU
?
•
Factorisable as
A
=
QR
or
B
=
QR
?
•
Invertible?
•
Normal?
•
Permutation?
•
Projection?
•
Selfadjoint?
•
Skewsymmetric?
•
Unitary?
Give reasons in each case!
[
Total: 10 marks
]
(2)
(a) Show that a unitary matrix
U
must have

det
(
U
)

=
1. Give an example of a unitary matrix whose
determinant is not 1.
[
2 marks
]
(b) If a diagonalisable matrix
A
has

det
(
A
)

=
1, must
A
be unitary?
[
1 mark
]
(c) Show that if
A
is skewsymmetric, then
A
+
I
is invertible.
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 Winter '10
 Ridout
 Math, Linear Algebra, Matrices, TA, Orthogonal matrix, Burnside Hall

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