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\headers{Math343}{Homework 2}{Due 3 Sep 2025}

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\noindent
\textbf{Directions:} You may work to solve these problems in groups, but all written work must be your own.  Show all work; no credit for solutions without work..

\begin{enumerate}

\item {}[1.1.38] Suppose $a$, $b$, $c$, and $d$ are constants such that $a$ is nonzero and the system below is consistent for all possible values of $f$ and $g$.  What can you say about the numbers $a$, $b$, $c$, and $d$?  Justify your answer.
\[\begin{array}{*{5}{r}}
	ax_1  &+&  bx_2  &=& f\\
	cx_1  &+&  dx_2  &=& g
\end{array}\]

\item {}[1.2.\{7-11\}] Find the general solutions of the systems whose augmented matrices are given below.
\begin{multicols}{2}
\begin{enumerate}
	\item $\left[
		\begin{array}{*{4}{r}}
		1 & 3 & 4 & 7\\
		3 & 9 & 7 & 6
		\end{array}\right]$
	\item $\left[
		\begin{array}{*{4}{r}}
		1 & 4 & 0 & 7\\
		2 & 7 & 0 & 11
		\end{array}\right]$
	\item $\left[
		\begin{array}{*{4}{r}}
		0 & 1 & -6 & 5\\
		1 & -2 & 7 & -4\\
		\end{array}\right]$
	\item $\left[
		\begin{array}{*{4}{r}}
		1 & -2 & -1 & 3\\
		3 & -6 & -2 & 2\\
		\end{array}\right]$
	\item $\left[
		\begin{array}{*{4}{r}}
		3 & -4 & 2 & 0\\
		-9 & 12 & -6 & 0\\
		-6 & 8  & -4 & 0
		\end{array}\right]$
	\item $\left[
		\begin{array}{*{5}{r}}
		1 & -7 & 0 & 6 & 5\\
		0 & 0 & 1 & -2 & -3\\
		-1 & 7 & -4 & 2 & 7
		\end{array}\right]$
\end{enumerate}
\end{multicols}

\item Determine a cubic polynomial $f(t) = a + bt + ct^2 + dt^3$ such that $f(-1)=-1$, $f(0)=0$, $f(1)=1$, and $f'(1)=0$.  Hint: use the four given pieces of information about $f$ to write four linear equations in the variables $a$, $b$, $c$, and $d$.

\end{enumerate}
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