Matrix Algebra - 1016 - 331
Fall 2005
Problem List & Set 2
Section Problems
1.1 2, 4, 5a, b, 9 – 14, 19 – 22
1.2 18, 19, 31, 33, 42 – 44, 46, 53, 54, 59 – 61, 64
2.1 35 – 37, 39 – 41, 44
2.2 9 – 12, 24 – 28, 35 – 38, 40 – 46, 49, 53, 54, 60
Set 2: Due Tuesday, September 20, at the start of class. Provide the necessary arguments for your conclusions.
2.1 36, 44
2.2 40, 46
24 First solve this problem as it is stated in the text. Then consider the following variation.
Suppose each entry must be a 0 or 1. How many different reduced row echelon
€
3×3
matrices
are there? Note the difference in the two questions being asked. In the textbook problem you
are asked for the number of different forms. In my problem you are asked for the actual
number of matrices that can have those forms.
A. In this problem we will set up a scheme, similar to one we looked at in class, for validating
€
3×1
vectors of a particular type. In our vectors the only permitted entries are 0, 1, 2, and 3. Let
€
z=
1
2
3
.
A vector v will be called valid if and only if
€
v⋅z≡0 mod 4
. For example,
€
v=
1
2
1
is valid, since
€
v⋅z=1⋅1+2⋅2+1⋅3=8≡0mod 4
, but
€
v=
2
1
1
is not valid, since
€
v⋅z=2⋅1+1⋅2+1⋅3=7≡3 mod 4
. I may return to this scheme in future problems, but here I will
ask just one brief question. How many valid vectors have the form
€
v=
1
a
b
?