Those matrices, as you say yourself, are very big and simply too big for one line. You could change the font size in the math environment, but this would be an inconsistency.
Another possibility would be to change the printable area (this is shown by showframe) using geometry. Click on »Open in Writelatex« just above the following code and uncomment the
\geometry
line.
Code: Select all
\documentclass[a4paper]{article}
\usepackage[utf8]{inputenc}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{showframe}
\usepackage{geometry}
%\geometry{margin=1cm}%Just for showing, these are bad margins!
\begin{document}
\[ f(x) =
\left \{
\begin{array} {rl}
\sin (\frac {1}{x}) & \mbox {dla} \ x \neq 0\\
0 & \mbox{dla } \ x \in \mathbb{R} \backslash \{0\}.
\end{array}
\right. \]
\[ f(x) = \begin{cases}
\sin (\frac {1}{x}) & \text {dla} \ x \neq 0\\
0 & \text{dla } \ x \in \mathbb{R} \backslash \{0\}.
\end{cases}
\]
\[
%\small
\footnotesize
\det \begin{pmatrix}
a_1,_1 & 0 & 0 & \ldots & 0 \\
a_2,_1 & a_2,_2 & 0 & \ldots & 0\\
a_3._1 & a_3,_2 & a_3,_3 & \ldots & 0\\
\vdots & \vdots & \vdots & \ddots & \vdots & \\
a_n-_1,_1 & \ldots & a_n-_1,_n-_1 & \ldots & a_n,_m
\end{pmatrix} = \prod _{i=1}^n a_{i,i}
\quad
D =\left[ \begin{array} {ccc |cccc}
a_1,_1 & \ldots & a_1,_n & & & \\
\vdots & \ddots & \vdots & & & \Theta\\
a_n,_1 & \ldots & a_n,_n & & & \\
\hline
& & & & b_1,_1 & \ldots & b_1,_n \\
& \Theta & & \vdots & \ddots & \vdots \\
& & & & b_n,_1 & \ldots & b_n,_n
\end{array}\right]
\]
\[
\det \begin{pmatrix}
a_1,_1 & 0 & 0 & \ldots & 0 \\
a_2,_1 & a_2,_2 & 0 & \ldots & 0\\
a_3._1 & a_3,_2 & a_3,_3 & \ldots & 0\\
\vdots & \vdots & \vdots & \ddots & \vdots & \\
a_n-_1,_1 & \ldots & a_n-_1,_n-_1 & \ldots & a_n,_m
\end{pmatrix} = \prod _{i=1}^n a_{i,i}
\quad
D =\left[ \begin{array} {ccc |cccc}
a_1,_1 & \ldots & a_1,_n & & & \\
\vdots & \ddots & \vdots & & & \Theta\\
a_n,_1 & \ldots & a_n,_n & & & \\
\hline
& & & & b_1,_1 & \ldots & b_1,_n \\
& \Theta & & \vdots & \ddots & \vdots \\
& & & & b_n,_1 & \ldots & b_n,_n
\end{array}\right]
\]
\end{document}
The smart way: Calm down and take a deep breath, read posts and provided links attentively, try to understand and ask if necessary.