How to write matrices and aligned equations in LaTeX

The matrix environments

amsmath has one matrix environment per bracket style. The content is always written the same way, only the delimiters change. All of them are used inside math mode, for example within \[ … \].

pmatrix, bmatrix and vmatrix in practice

Write the entries row by row. Separate columns with & and end every row except the last with \\. A vmatrix is the usual way to write a determinant.

\begin{pmatrix}
  a & b \\
  c & d
\end{pmatrix}

\begin{bmatrix}
  1 & 0 & 0 \\
  0 & 1 & 0 \\
  0 & 0 & 1
\end{bmatrix}

\det(A) = \begin{vmatrix}
  a & b \\
  c & d
\end{vmatrix} = ad - bc

Large matrices with dots

For general m × n matrices, \cdots draws centered horizontal dots, \vdots vertical dots and \ddots diagonal dots. By default amsmath matrices allow up to 10 columns; for more, set \setcounter{MaxMatrixCols}{20} in the preamble.

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}

Augmented matrices and small inline matrices

The matrix environments cannot draw a vertical line. For an augmented coefficient matrix, use the array environment with a column specification such as {cc|c} and wrap it in \left( … \right). Each letter stands for a centered column and | draws the line.

Inside running text, a normal pmatrix is too tall. Use smallmatrix and add the brackets yourself, for example $\bigl(\begin{smallmatrix} a & b \\ c & d \end{smallmatrix}\bigr)$.

\left(\begin{array}{cc|c}
  1 & 2 & 5 \\
  3 & 4 & 6
\end{array}\right)

Aligning equations with align

align puts several equations below each other and aligns them at the & – usually placed directly before the equals sign. Every line gets a number; align* leaves out the numbers, and \notag removes the number from a single line.

align is a display environment of its own. Do not put it inside \[ … \] or equation; that causes an error. If you need an aligned block inside an existing display, use aligned instead.

\begin{align}
  f(x) &= (x + 1)^2 \\
       &= x^2 + 2x + 1
\end{align}

\begin{align*}
  a &= b + c \\
  d &= e - f
\end{align*}

\[
  \begin{aligned}
    (a + b)^2 &= (a + b)(a + b) \\
              &= a^2 + 2ab + b^2
  \end{aligned}
\]

Case distinctions with cases

cases sets a left brace with two columns: the value and the condition, separated by &. \text{…} keeps words upright and with normal spacing.

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

Systems of equations

For a system of linear equations, combine aligned with a left brace. \right. is the invisible partner that \left\{ needs. Thanks to the &, the equals signs line up neatly. If you prefer the matrix form, write the system as an augmented matrix as shown above.

\left\{
\begin{aligned}
  2x + 3y &= 7 \\
   x -  y &= 1
\end{aligned}
\right.

Common errors

• “Misplaced alignment tab character &”: an & outside a matrix, align or cases environment, or a typo in the environment name.
• “Environment pmatrix undefined”: amsmath is not loaded.
• Every \begin{…} needs an \end{…} with exactly the same name.
• No \\ after the last row – it can add an empty row or extra space.
• A missing \right. after \left\{ stops the formula from rendering.

Writing matrices on iPhone and iPad

Typing & and \\ on a software keyboard is tedious. Texlet has building blocks for matrices and environments above the keyboard, and pressing Return after \begin{…} adds the matching \end{…}, which avoids the most common error on this list. The live preview typesets matrices and cases as you type, also offline, so you see right away whether a row is missing an entry. With Texlet Pro you can export the result as PDF or PNG or copy it as an image.