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Problems B and F are graded.
Problem B: Find an eigenvector corresponding to the eigenvalue $\lambda=4$ of the matrix
$$A=\begin{bmatrix} 3&0&-1 \\ 2&3&1 \\ -3&4&5 \end{bmatrix}.$$
Solution: We must find a vector $\vec{x}$ such that $A\vec{x}=4\vec{x},$ i.e.
$$\begin{bmatrix} 3&0&-1 \\ 2&3&1 \\ -3&4&5 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} = 4 \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}.$$
or in other words
$$\begin{bmatrix} 3x_1 - x_3 \\ 2x_1+3x_2+x_3 \\ -3x_1+4x_2+5x_3 \end{bmatrix} = \begin{bmatrix} 4x_1 \\ 4x_2 \\ 4x_3 \end{bmatrix}.$$
Subtracting the right-hand-side from both sides of the equation yields
$$\begin{bmatrix} -x_1-x_3 \\ 3x_1-x_2+x_3 \\ -3x_1 + 4x_2 + x_3 \end{bmatrix} = \vec{0},$$
yielding the system of equations
$$\left\{ \begin{array}{ll}
-x_1-x_3 &= 0 \\
3x_1-x_2+x_3 &= 0 \\
-3x_1 + 4x_2 + x_3 &= 0.
\end{array} \right.$$
The reduced echelon form of the assocaited augmented matrix is
$$\begin{bmatrix}
1 & 0 & 1&0 \\
0&1&1&0 \\
0&0&0&0
\end{bmatrix}.$$
which is equivalent to the system
$$\left\{ \begin{array}{ll}
x_1 + x_3 &= 0 \\
x_2 + x_3 &= 0
\end{array} \right.$$
and so we see that $x_1=-x_3$ and $x_2=-x_3$ with free variable $x_3$. Thus we see that an eigenvector $\vec{x}$ is any vector of the form
$$\vec{x} = \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} = \begin{bmatrix} -x_3 \\ -x_3 \\ x_3 \end{bmatrix} = x_3\begin{bmatrix} -1 \\ -1 \\ 1 \end{bmatrix}.$$
So pick the simplest constant, $x_3=1$, to get an eigenvector $\begin{bmatrix} -1 \\ -1 \\ 1 \end{bmatrix}$.
Problem F: Use integration by parts to calculate the antiderivative of $f(x)=\log(x)$. (Hint: use $u=\log x$ and $dv=1$. Also recall that $\dfrac{d}{dx} \log x = \dfrac{1}{x}$)
Solution: Using $u=\log x$ yields $du = \dfrac{1}{x} dx$ and using $dv = 1 dx$ yields $v=x$. Therefore by integration by parts,
$$\displaystyle\int \log(x) dx = \displaystyle\int u dv = uv - \displaystyle\int v du = x\log x - \displaystyle\int 1 dx = x \log x - \displaystyle\int x \dfrac{1}{x} dx = x\log x - x + C,$$
where $C$ is a constant introduced when finding the anti-derivative of $-1$.
