WeBWorK Standalone Renderer

Assuming $a, b$ and $k$ are constants, calculate the following derivative.

$\displaystyle \frac{d}{dt} \left( \left\lbrack \begin{array}{c} a \\ b \end{array} \right\rbrack e^{k t} \right) =$

$\left.\vphantom{\begin{array}{c}\!\strut\\\!\strut\\\!\strut\\\end{array}}\right[$		$\left.\vphantom{\begin{array}{c}\!\strut\\\!\strut\\\!\strut\\\end{array}}\right]$

Find a value of $k$ so that $\displaystyle \left\lbrack \begin{array}{r} 1 \\ 1 \end{array} \right\rbrack e^{k t}$ is a solution to $\displaystyle \boldsymbol{\vec{x}^{\,\prime}} = \left\lbrack \begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array} \right\rbrack \boldsymbol{\vec{x}}.$
$k =$

Find a value of $k$ so that $\displaystyle \left\lbrack \begin{array}{r} 1 \\ -1 \end{array} \right\rbrack e^{k t}$ is a solution to $\displaystyle \boldsymbol{\vec{x}^{\,\prime}} = \left\lbrack \begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array} \right\rbrack \boldsymbol{\vec{x}}.$
$k =$

Write down the general solution in the form $x_1(t) = ?$ and $x_2(t) = ?$ , i.e., write down a formula for each component of the solution. Use $A$ and $B$ to denote arbitrary constants.
$x_1(t) =$
$x_2(t) =$