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\begin{document}

\title{Chapter Check for Chapters 3 and 4}
\maketitle

1. Consider the matrix
\[
A=\left(
\begin{array}
[c]{ccccc}%
1 & 2 & -1 & 3 & 1\\
2 & 1 & 2 & -1 & 1\\
1 & 0 & 0 & 1 & 1
\end{array}
\right)  .
\]
\bigskip

\qquad a. Using row and column operations, find invertible matrices $P$ and
$Q$ such that $PAQ$ is of the block form
\[
\left(
\begin{array}
[c]{cc}%
I & 0\\
0 & 0
\end{array}
\right)
\]
for a certain size identity matrix $I$ where the other matrices have all zero entries.

\qquad b. Put $A$ in reduced row echelon form.

\qquad c. Use the reduced row echelon form of $A$ to find a collection of
columns of $A$ that form a basis for the column space.\bigskip



2. a. Compute the determinant of the matrix
\[
\left(
\begin{array}
[c]{cccc}%
1 & 2 & 3 & 4\\
2 & 6 & 12 & 9\\
3 & 6 & 10 & 15\\
4 & 8 & 12 & 14
\end{array}
\right)
\]
by using row operations (Hint: Recall the determinant of an upper triangular matrix).

\qquad b. Recall a matrix $A\in F^{n\times n}$ is skew-symmetric if
$A^{T}=-A.$ If the field is not of characteristic 2, use the determinant to
show that $A$ has rank less than $n$ if $n$ is odd.

\bigskip

3. (Comprehensive/graduate option only) Let $A\in F^{m\times n}$ and $B\in
F^{n\times m}.$ Show that $\det\left(  I_{m}+AB\right)  =\det\left(
I_{n}+BA\right)  $ by showing that
\[
\left(
\begin{array}
[c]{cc}%
I_{m} & -A\\
B & I_{n}%
\end{array}
\right)  =\left(
\begin{array}
[c]{cc}%
I_{m} & 0\\
B & I_{n}%
\end{array}
\right)  \left(
\begin{array}
[c]{cc}%
I_{m} & -A\\
0 & I_{n}+AB
\end{array}
\right)  =\left(
\begin{array}
[c]{cc}%
I_{m}+BA & -A\\
0 & I_{n}%
\end{array}
\right)  \left(
\begin{array}
[c]{cc}%
I_{m} & 0\\
B & I_{n}%
\end{array}
\right)
\]
and using properties of determinants. [This is called Sylvester's determinant theorem.]


\end{document}