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# Math 371 Problem Set

**Diagonalization/Rank of a matrix/Linear systems**

1) For
find elementary transformation

such that
has
the canonical form given in the class
with
D a diagonal matrix and

corresponding
zero matrices, in the following cases:

2) Let
and

and considerProve
the following:

a) Suppose that for some
one has det
Then for all

there exist
such that det

b) rank(A) is the maximum over all the k such that det for some

3) Analyze which of the following matrices are diagonalizable, where the
coefficients are in an arbitrary field,

and if so, find the diagonal form:

4) Discuss the following systems of linear equations, where the coefficients
are from an arbitrary commutative

ring R with 1_{R}:

**Algebraic field extensions**

• Problems 4, 5, 6, 8, 9, at the end of Section 5.1, Ch.5, of Herstein’s book Topics in Algebra.