Matrices, cases and aligned equations in PDF

# Math structures

## Matrices

$ A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}, \qquad \det A = \begin{vmatrix} a & b \\ c & d \end{vmatrix} = ad - bc, \qquad A^{-1} = \frac{1}{ad - bc} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix} $

$ R_z(\theta) = \begin{bmatrix} \cos\theta & -\sin\theta & 0 \\ \sin\theta & \cos\theta & 0 \\ 0 & 0 & 1 \end{bmatrix} \qquad A = \begin{pmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{pmatrix} $

A small matrix inline, $\left(\begin{smallmatrix} 1 & 0 \\ 0 & 1 \end{smallmatrix}\right)$, keeps the line height.

## Piecewise definitions

$ |x| = \begin{cases} x & \text{if } x \geq 0 \\ -x & \text{otherwise} \end{cases} \qquad f(n) = \begin{cases} n/2 & n \equiv 0 \pmod{2} \\ 3n + 1 & n \equiv 1 \pmod{2} \end{cases} $

$ \operatorname{sgn}(x) = \begin{dcases} \dfrac{x}{|x|} & x \neq 0 \\ 0 & x = 0 \end{dcases} $

## Aligned derivations

$ \begin{aligned} (x + y)^3 &= (x + y)(x + y)^2 \\ &= (x + y)(x^2 + 2xy + y^2) \\ &= x^3 + 3x^2 y + 3x y^2 + y^3 \end{aligned} $

$ \begin{aligned} \int_0^{\pi} \sin^2 x \,\mathrm{d}x &= \int_0^{\pi} \frac{1 - \cos 2x}{2} \,\mathrm{d}x \\ &= \frac{\pi}{2} - \frac{1}{4}\Big[\sin 2x\Big]_0^{\pi} = \frac{\pi}{2} \end{aligned} $

Systems of equations, with a phantom to align the signs:

$ \begin{aligned} 2x + 3y - z &= \phantom{-}5 \\ x - y + 4z &= -2 \\ 3x + 2y + 2z &= \phantom{-}9 \end{aligned} \qquad \Longrightarrow \qquad \begin{pmatrix} 2 & 3 & -1 \\ 1 & -1 & 4 \\ 3 & 2 & 2 \end{pmatrix} \begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} 5 \\ -2 \\ 9 \end{pmatrix} $

## Ruled arrays

$ \begin{array}{|l|c|r|} \hline \text{Grade} & \text{Count} & \text{Share} \\ \hline \text{A} & 12 & 24\,\% \\ \text{B} & 23 & 46\,\% \\ \text{C} & 15 & 30\,\% \\ \hline \end{array} \qquad \begin{array}{c|cc} & 0 & 1 \\ \hline 0 & 0 & 1 \\ 1 & 1 & 0 \end{array} $

## Growing delimiters

$ \left( \frac{\frac{a}{b}}{\frac{c}{d}} \right) \qquad \left[ \sum_{i=1}^{n} \frac{1}{i^2} \right]^{2} \qquad \left\{ x \in \mathbb{R} \;\middle|\; x^2 < 2 \right\} \qquad \left. \frac{\mathrm{d}y}{\mathrm{d}x} \right|_{x = 0} \qquad \left\langle \psi \middle| \phi \right\rangle $

$ \Bigg\langle \left\{ \left[ \left( \frac{\frac{\frac{a}{b}}{c}}{d} \right) \right] \right\} \Bigg\rangle \qquad \big( x \big) \; \Big( x \Big) \; \bigg( x \bigg) \; \Bigg( x \Bigg) $

## Braces, stacks and boxes

$ \underbrace{1 + 2 + \cdots + n}_{n \text{ terms}} = \frac{n(n+1)}{2} \qquad \overbrace{x \cdot x \cdots x}^{k \text{ factors}} = x^k \qquad a \overset{\text{def}}{=} b \qquad A \stackrel{f}{\longrightarrow} B $

$ \boxed{E = mc^2} \qquad \boxed{\int_{-\infty}^{\infty} e^{-x^2}\,\mathrm{d}x = \sqrt{\pi}} \qquad {\color{teal} \text{colored}} \quad \textcolor{red}{x} + \textcolor{blue}{y} $