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\ihead{Math 201-003 Fall 2019 (Darij Grinberg): homework set 3}
\ohead{page \thepage}
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\begin{document}

\begin{center}
\textbf{Math 201-003 Fall 2019 (Darij Grinberg): homework set 3}

due date: Wednesday 2019-11-11 at the beginning of class, or earlier by Blackboard
\end{center}

\begin{exercise}
Prove that
\[
\det\left(  AB\right)  =\det A\cdot\det B
\]
holds for any $2\times2$-matrices $A=\left(
\begin{array}
[c]{cc}%
a & b\\
a^{\prime} & b^{\prime}%
\end{array}
\right)  $ and $B=\left(
\begin{array}
[c]{cc}%
c & d\\
c^{\prime} & d^{\prime}%
\end{array}
\right)  $.

[In class, I said that this holds for square matrices of any size, but I
didn't give a proof. You cannot use this general fact without proof here.]
\end{exercise}

\begin{exercise}
\label{exe.ps4.5}Let $a,b,c,d,e,f,g,h,i,j,k,\ell,m,n,o,p$ be any numbers.

\textbf{(a)} Find a simple formula for the determinant%
\[
\det\left(
\begin{array}
[c]{cccc}%
a & b & c & d\\
\ell & 0 & 0 & e\\
k & 0 & 0 & f\\
j & i & h & g
\end{array}
\right)  .
\]


\textbf{(b)} Find a simple formula for the determinant%
\[
\det\left(
\begin{array}
[c]{ccccc}%
a & b & c & d & e\\
f & 0 & 0 & 0 & g\\
h & 0 & 0 & 0 & i\\
j & 0 & 0 & 0 & k\\
\ell & m & n & o & p
\end{array}
\right)  .
\]
(Do not mistake the \textquotedblleft$o$\textquotedblright\ for a
\textquotedblleft$0$\textquotedblright.)

[\textbf{Hint:} Part \textbf{(b)} is simpler than part \textbf{(a)}.]
\end{exercise}

\begin{exercise}
Find%
\[
\det\left(
\begin{array}
[c]{ccccc}%
5 & 4 & 3 & 2 & 1\\
1 & 0 & 0 & 3 & 0\\
2 & 0 & 0 & 0 & 0\\
2 & 2 & 4 & 3 & 0\\
-3 & 0 & 1 & 2 & 0
\end{array}
\right)  .
\]

\end{exercise}

\begin{exercise}
Here is a $5\times5$-matrix: $\left(
\begin{array}
[c]{ccccc}%
0 & 1 & 0 & 0 & 0\\
0 & 0 & 4 & 0 & 0\\
0 & 0 & 1 & 2 & 0\\
0 & 0 & 3 & 0 & 0\\
1 & 0 & 2 & 2 & 1
\end{array}
\right)  $. Its determinant is $0$.

\vspace{0.5pc}

Find a $0$ entry which can be replaced by a $1$ to give a nonzero determinant.
(You can box this entry in the matrix. Note that you cannot replace more than
one entry simultaneously.)
\end{exercise}


\end{document}