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\section{Matrices}

\subsection{Example 1}

$$
\begin{pmatrix}
    a_{11} & a_{12} & \cdots & a_{1n} \\
    a_{21} & a_{22} & \cdots & a_{2n} \\
    \vdots & \vdots & \ddots & \vdots \\
    a_{n1} & a_{n2} & \cdots & a_{nn}
\end{pmatrix}
$$

\subsection{Example 2}

\begin{align*}
    \begin{vmatrix} 1 & 1 & 0 \\ 0 & 1 & 1 \\ 1 & 0 & 1 \end{vmatrix}
    &= 1\cdot \begin{vmatrix} 1 & 1 \\ 0 & 1 \end{vmatrix} - 0\cdot \begin{vmatrix} 1 & 0 \\ 0 & 1 \end{vmatrix}
    + 1\cdot \begin{vmatrix} 1 & 0 \\ 1 & 1 \end{vmatrix}\\
    &= 1\cdot 1 + 1\cdot 1\\
    &= 2.
\end{align*}

\subsection{Example 3}

Consider the matrix $\alpha = \left(\begin{smallmatrix} 1 & 1 \\ 0 & 1\end{smallmatrix}\right)$.  Observe that for any integer $n$ we have
$$\alpha^n = \begin{pmatrix} 1 & n \\ 0 & 1 \end{pmatrix}.$$

\section{Aligning Equations}

Play with the following examples and create your own.

\subsection{Example 1}

\begin{multline}
    \sum_{m=1}^{\infty}\frac{c_m}{m^{\nu}}\sqrt{\left(\frac{2}{m\pi x}\right)} \Bigg\{ \cos(mx-\tfrac{1}{2}\nu\pi-\tfrac{1}{4}\pi-\eta_m)\\
           -\frac{4\nu^2-1}{8mx}\sin(mx-\tfrac{1}{2}\nu\pi-\tfrac{1}{4}\pi-\eta_m) \Bigg\}+\frac{b_m}{m^{\nu}}
           \frac{(\tfrac{1}{2}mx)^{\nu-1}}{\Gamma(\nu+\frac{1}{2})\Gamma(\frac{1}{2})}
\end{multline}

\subsection{Example 2}

\begin{multline}
    \sum_{m,n=0}^{\infty} e^{-a^n b^m x} = \frac{\log^2 x}{2\log a\log b}+\log x\left(
    \frac{\gamma}{\log a\log b}-\frac{1}{2\log a}-\frac{1}{2\log b}\right) \\
    \shoveright{+\frac{1}{12}\left(\frac{\log a}{\log b}+\frac{\log a}{\log b}+\frac{\pi^2+6\gamma^2}{\log a\log b}\right)
    -\frac{\gamma}{2}\left(\frac{1}{\log a}+\frac{1}{\log b}\right)+\frac{1}{4}} \\
    \shoveright{+\sum_{n=1}^{\infty} \frac{(-1)^n}{n!(a^n-1)(b^n-1)} +\frac{1}{\log a}\sum_{\substack{n=-\infty\\ n\neq 0}}^{\infty}
    \frac{\Gamma\left(-\frac{2n\pi i}{\log a}\right) x^{2n\pi i/\log a}}{1-b^{2n\pi i/\log a}} }\\
    +\frac{1}{\log b} \sum_{\substack{n=-\infty\\ n\neq 0}}^{\infty}
    \frac{\Gamma\left(-\frac{2n\pi i}{\log b}\right) x^{2n\pi i/\log b}}{1-a^{2n\pi i/\log b}}.
\end{multline}



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