3201: COMMUTATIVE ALGEBRA
FALL 2010. EXERCISE SHEET 7
TO BE HANDED IN ON DECEMBER 8TH
Problems marked with a star (∗)will be assessed and count towards the final
mark. Problems marked with an (O) are optional, you are strongly advised to try
these. They will be marked for your own convenience, but the mark will not count
towards your final mark. Problems marked with a (C) are challenge questions.
Exercise 1 (*).Find the Smith Normal Form of the following matrices:
i)
10 2
4 4
12 3
8 3
in M4×2(Z).
ii)
15 0 0
0 3 0
0 0 5
in M3(Z).
iii)
5325
2−2 0 −2
4420
in M3×4(Z).
iv)
(x3−1) 0 0
0 (x2−1) 0
0 0 x
in M3(Q[x]).
v)
x1 + x2
5x21−x
1 + x x2
in M3×2(Q[x]).
Exercise 2 (*).For the matrix A=
10 2
4 4
12 3
8 3
as in part i) of the previous question,
find the matrices X∈GL(4,Z)and Y∈GL(2,Z)such that XAY is the Smith
Normal form of A. Check your answer showing that the product is actually the
Smith normal form that you found before.
Hint: The matrix Xkeeps track of the row transformations effected on A, whilst Y
keeps track of the column transformations.
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