Matrices and Piecewise Functions in LaTeX: pmatrix, cases

Grids built from & and \\: matrices, determinants, column vectors and functions in several parts. About twelve minutes.

A matrix is a grid, and LaTeX builds grids from two characters: & moves to the next column and \\ starts the next row. The same two build functions defined in several parts.

A first matrix

A matrix goes between \begin{pmatrix} and \end{pmatrix}. Inside, separate the entries in a row with & and end each row with \\:

\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}

Edit the LaTeX

The last row needs no \\. Spaces and line breaks inside are ignored, so you can lay the source out like the matrix itself:

LaTeX

\begin{pmatrix}
  1 & 2 \\
  3 & 4
\end{pmatrix}

The brackets round it

The first letters of the name choose the brackets:

matrix: none.
pmatrix: round brackets, $( \; )$ .
bmatrix: square brackets, $[ \; ]$ .
Bmatrix: curly braces, $\{ \; \}$ .
vmatrix: straight bars, for a determinant.
Vmatrix: double bars, for a norm.

\begin{vmatrix} a & b \\ c & d \end{vmatrix} = ad - bc

Edit the LaTeX

Column vectors

A column vector is a matrix with one column: no &, just \\ between the rows.

\mathbf{v} = \begin{pmatrix} x \\ y \\ z \end{pmatrix}

Edit the LaTeX

Large matrices

For a matrix of any size, fill the gaps with dots: \cdots along a row, \vdots down a column, and \ddots along the diagonal.

\begin{pmatrix} a_{11} & \cdots & a_{1n} \\ \vdots & \ddots & \vdots \\ a_{m1} & \cdots & a_{mn} \end{pmatrix}

Edit the LaTeX

A transpose and an inverse are powers: A^T gives $A^T$ and A^{-1} gives $A^{-1}$ .

Functions in several parts

cases makes the big brace of a function defined piece by piece. Each line is the value, then &, then the condition, and \\ ends the line. \text{...} writes words, as you saw in lesson 2.

|x| = \begin{cases} x & \text{if } x \geq 0 \\ -x & \text{if } x < 0 \end{cases}

Edit the LaTeX

cases also writes a system of equations: one equation on each line.

\begin{cases} x + y = 3 \\ x - y = 1 \end{cases}

Edit the LaTeX

Practise

$\begin{pmatrix} a & b \\ c & d \end{pmatrix}$
$\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$
$\det \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} = -2$
$\begin{pmatrix} x \\ y \end{pmatrix}$
$A^{-1} = \frac{1}{ad - bc} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$
$\begin{pmatrix} a_{11} & \cdots & a_{1n} \\ \vdots & \ddots & \vdots \\ a_{m1} & \cdots & a_{mn} \end{pmatrix}$
$f(x) = \begin{cases} 1 & x > 0 \\ 0 & x = 0 \\ -1 & x < 0 \end{cases}$
$\begin{cases} x + y = 3 \\ x - y = 1 \end{cases}$

In short

& moves to the next column, \\ to the next row.
pmatrix, bmatrix, Bmatrix, vmatrix and Vmatrix choose the brackets.
\cdots, \vdots and \ddots fill large matrices; cases builds functions in parts and systems of equations.

Next, in lesson 10: lining up equations over several lines.

Matrices and cases

A first matrix

The brackets round it

Column vectors

Large matrices

Functions in several parts

Practise

In short

Would you rather write it by hand?