\documentclass{exam}

\begin{document}

\begin{center}
\textbf{\Large MATH 3912 - Assignment 1}
\end{center}
\normalsize

\begin{questions}

\question
Compute the determinant of
$$
\left(
\begin{array}{ccc}
6 & 0 & -8 \\
8 & 3 &  0 \\
6 & 3 & -4 \\
\end{array}
\right)
$$

\vfill
\question
Compute the determinant of the lower triangular matrix
$$
\left(
\begin{array}{cccc}
a_{11} & 0      & 0      & 0 \\
a_{21} & a_{22} & 0      & 0 \\
a_{31} & a_{32} & a_{33} & 0 \\
a_{41} & a_{42} & a_{43} & a_{44}
\end{array}
\right)
$$

\vfill
\question
Suppose $A$ is a lower triangular $n$ by $n$ matrix as defined below.
Show that $\det(A) = \prod_{i=1}^n a_{ii}$, or in other words the determinant 
of $A$ is the product of the entries in the main diagonal.
$$
A =
\left(
\begin{array}{cccccc}
a_{11} & 0      & \dots  &        &        & 0      \\
a_{21} & a_{22} & 0      & \dots  &        & 0      \\
a_{31} & a_{32} & a_{33} & 0      &\dots   & 0      \\
\vdots &        &        & \ddots & \ddots &\vdots \\
a_{n1} & a_{n2} & a_{n3} & \dots  &        & a_{nn}
\end{array}
\right)
$$

\vfill
\question
Find the determinant of the matrix $V$ defined as follows:

$$
V = 
\left(
\begin{array}{ccc}
1 & z_0 & z_0^2 \\
1 & z_1 & z_1^2 \\
1 & z_2 & z_2^2 \\
\end{array}
\right)
$$

\vfill
\question
Solve the following system of equations, if there is a solution:

\begin{eqnarray*}
 8 x_1 - 2 x_2 +  x_3  &=& 1 \\
 2 x_1 - 8 x_2         &=& 7 \\
-2 x_1 - 6 x_2 - 7 x_3 &=& 2 \\
\end{eqnarray*}

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\question
Suppose $p(x) = a_0 + a_1 x + a_2 x^2$ is a quadratic polynomial. 
Find the coefficients of that polynomial so that

\begin{eqnarray*}
p(-1) &=& 1 \\
 p(0) &=& 4 \\
 p(1) &=& 1 \\
\end{eqnarray*}

\vfill
\question
Suppose 
$$
\vec v_1 = \left<7, -5, 5\right> \hbox{, }
\vec v_2 = \left<6, 0, 0 \right> \hbox{, and }
\vec v_3 = \left<-3, 5, 0\right>
$$
 are three vector in $R^3$.
Are they linearily independent? What about the vectors
$$
\vec w_1 = \left<-3, -6,  -7\right> \hbox{, }
\vec w_2 = \left< 0,  2, -15\right> \hbox{, and }
\vec w_3 = \left< 3,  8, -8 \right>
$$
How about
$$
\vec z_1 = \left<6, 0, 0 \right> \hbox{, }
\vec z_2 = \left<-3, 5, 0\right> \hbox{, }
\vec z_3 = \left<-3, -6,  -7\right> \hbox{, and }
\vec z_4 = \left< 0,  2, -15\right>
$$


\vfill
\end{questions}
\end{document}
