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\begin{document}

\noindent
{\large \bf Practice Problems.}

\bigskip

\begin{enumerate}
\item[(1)] (Berkeley, Prelim, S00) 
Are the $4\times 4$ matrices 
\[
A = \left[\begin{array}{cccc}
          1&0&0&0\\
          0&-1&0&0\\
          0&0&0&1\\
          0&0&0&0
          \end{array}\right],\;\mbox{\rm and}\;B = 
\left[\begin{array}{cccc}
          -1&0&0&0\\
          -1&1&1&-1\\
          -1&0&0&1\\
          -1&0&1&0
          \end{array}\right]\]
similar? Explain your reasoning. 
\bigskip
\item[(2)] (Berkeley, Prelim, S00) 
 Let $A$ be an $n\times n$ matrix over ${\BF}$
whose minimal polynomial has degree $k$.
Prove that if $\lambda\in {\BF}$ is not an eigenvalue of $A$, 
then there is a polynomial $p(x)\in {\BF}[x]$ of degree $<k$ 
such that $p(A) = (A-\lambda I)^{-1}$.
\bigskip
\item[(3)] (Berkeley, Prelim, S99) 
Let $M$ be an $n\times n$ matrix with 
entries in the polynomial ring ${\BR}[t]$ such that 
\[M^3 = \left[\begin{array}{ccc}
          t&0&0\\
          0&t&0\\
          0&0&t\\
          \end{array}\right].\]
Let $N$ be the matrix with real entries obtained from $M$ by 
substituting $0$ in for $t$. Prove that $N$ is similar to
\[\left[\begin{array}{ccc}
          0&1&0\\
          0&0&1\\
          0&0&0\\
          \end{array}\right].\]
\bigskip
\item[(4)] (Berkeley, Prelim, S98) 
Let $A$ be an $N\times N$ complex matrix whose
eigenvalues sum to $0$. Show that $A$ 
is similar to a matrix with all $0$'s along the main diagonal. 
\bigskip
\item[(5)] (Berkeley, Prelim, F97) 
Prove that if $A$ is a $2\times 2$ matrix over the 
integers such that $A^n = I$ for some strictly positive integer $n$, 
then $A^{12} = I$.
\bigskip
\item[(6)] (Berkeley, Prelim, F94) 
Write down a list of $5\times 5$ complex matrices, 
as long as possible, with the following properties: 
\begin{enumerate}
\item[(a)] The characteristic polynomial of each matrix in the list is $x^5$; 
\item[(b)] The minimal polynomial of each matrix in the list is $x^3$; 
\item[(c)] No two matrices in the list are similar. 
\end{enumerate}
\bigskip
\item[(7)] (Dartmouth, Quals, 96) 
If $G$ is a finite abelian group, and $G\otimes {\BZ}_p = 0$
for all primes $p$, show that $G = 0$. Does the result
remain true if $G$ is infinite?
\bigskip
\item[(8)] (Dartmouth, Quals, 96) 
Show that if $T:V\rightarrow V$ is a linear transformation 
over a field of characteristic $\neq 2$, and $T^2 = I$,
then $T$ is diagonalizable.
\bigskip
\item[(9)] (Missouri at Columbia, Quals, F00) 
Exhibit all $6\times 6$ nilpotent complex matrices, up
to similarity. (A matrix is {\it nilpotent} if some power is $0$.)
\bigskip
\item[(10)] (Nebraska, Quals, S00) 
Let $A$ be an $8\times 8$ real matrix with determinant
$16$ and minimal polynomial $(x-2)^2(x^2+1)$. What are
the possible rational canonical forms and Jordan canonical
forms for $A$?
\bigskip
\item[(11)] (Penn State, Quals, F99)
Determine the structure of the abelian group
given by generators $x, y$, and $z$ and the relations
$7x+2y+3z=0$, $21x+8y+9z=0$, and $5x-4y+3z=0$.
\bigskip
\item[(12)] (Temple, Quals, F95)
Let $T: V\rightarrow V$ be an endomorphism. Suppose 
that for some $k\geq 1$ and some $v\in V$ we have
$T^k(v) = 0$ and $T^{k-1}(v)\neq 0$. Show
that $x^k$ divides the minimal polynomial of $T$.
\bigskip
\item[(13)] (Columbia, Quals, F95)
Let $M$ be a Noetherian $\m r$-module, and let
$\varphi: M\rightarrow M$ be a surjective 
homomorphism. Show that $\varphi$ is an isomorphism.
\bigskip
\item[(14)] (Johns Hopkins, Quals, F00)
Let $\m r = {\BQ}[x]/(x^7-1)$. Give an example
of an $\m r$-module that is not free
but is a direct summand of a free $\m r$-module.
\bigskip
\item[(15)] (Rochester, Prelims, S95)
Show that an Artinian ring has only finitely
many maximal ideals.
\bigskip
\item[(16)] (Oklahoma, Quals, S00)
Show that if $\m r[x]$ is a PID, then
$\m r$ is a field.
\end{enumerate}
\end{document}

