2 (b) There are a lot of 2 x 2 matrices with entries in Z/3 and probably you don't want to list all of them. It turns out that there is a different way to figure out how many elements are in GL(2, Z/3). You might remember that a 2 x 2 matrix with entries in R is invertible if the first column is not the zero vector and if the second column is not in the span of the first column, which is to say, not a multiple of the first column. The same works for matrices with entries in Z/p (take this on faith). Your job is to find the number of elements in GL(2,Z/3). You first need to figure out the number of nonzero column vectors (two entries long) with entries in Z/3. Then, given a nonzero vector, you need to figure out how many vectors are not multiples of it.
2 (b) There are a lot of 2 x 2 matrices with entries in Z/3 and probably you don't want to list all of them. It turns out that there is a different way to figure out how many elements are in GL(2, Z/3). You might remember that a 2 x 2 matrix with entries in R is invertible if the first column is not the zero vector and if the second column is not in the span of the first column, which is to say, not a multiple of the first column. The same works for matrices with entries in Z/p (take this on faith). Your job is to find the number of elements in GL(2,Z/3). You first need to figure out the number of nonzero column vectors (two entries long) with entries in Z/3. Then, given a nonzero vector, you need to figure out how many vectors are not multiples of it.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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I need help with part b
Transcribed Image Text:2
(b)
There are a lot of 2 x 2 matrices with entries in Z/3 and probably you don't want
to list all of them. It turns out that there is a different way to figure out how many elements
are in GL(2, Z/3). You might remember that a 2 x 2 matrix with entries in R is invertible if
the first column is not the zero vector and if the second column is not in the span of the first
column, which is to say, not a multiple of the first column. The same works for matrices with
entries in Z/p (take this on faith). Your job is to find the number of elements in GL(2, Z/3).
You first need to figure out the number of nonzero column vectors (two entries long) with
entries in Z/3. Then, given a nonzero vector, you need to figure out how many vectors are
not multiples of it.
(c) If you succeed at (b), you probably can give me a formula for the number of
elements of GL(2, Z/p) for any prime p. Try it!
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