This page illustrates "Mathjax" which can integrate mathematics into web pages. Below is some miscellaneous mathematics. Note that the mathematics scales.. This is a web page, not a pdf file, it automatically changes adjusts the width of the page if you change the width of the web page. One can also set a zoom feature on the mathematics. To see this point at a math formula and hold down the control key (on a Mac) and adjust the settings.

When $a \ne 0$ there are two solutions to \(ax^2 + bx + c = 0\) and they are $$x = {-b \pm \sqrt{b^2-4ac} \over 2a}.$$ \begin{equation*} \psi=1-\phi=-\frac{1}{\phi}=\frac{1-\sqrt{5}}{2}=-0.61803989\ldots. \end{equation*} $$ A\Big( \begin{matrix} i & j & k\\ t &u &v\\ \end{matrix}\Big), $$

Theorem 1. Let $\Delta_A(\lambda)$ be the characteristic polynomial of the matrix $A$, that is \begin{equation} \Delta_A(\lambda)=\det(\lambda I-A), \end{equation} then \begin{equation} \Delta_A(\lambda)=\sum_{k=0}^n (-1)^{n-k}\lambda^k \sigma_{n-k}(A)= \sum_{k=0}^n (-1)^{k}\lambda^{n-k }\sigma_{k}(A). \label{eq:characteristic} \end{equation} where $\sigma_k(A)= $ the sum of all the principal minors of order $k$ of $A$, with $\sigma_0(A)=1$.

proof \begin{eqnarray*} \Delta(\lambda )&=&\det[\lambda I-A]\\ &=&\sum_{k=0}^n(-1)^{n-k}\lambda^k\det[A : I]_{T_k}\\ &=&\sum_{k=0}^n(-1)^{n-k}\lambda ^k\sigma_{n-k}(A), \end{eqnarray*} \begin{align} &\begin{bmatrix} 1&2&1\\2&4&2\\1&2&1 \end{bmatrix} \begin{bmatrix} x_1\\x_2\\x_3 \end{bmatrix}\\ &=(x_1+2x_2+x_3)\begin{bmatrix} 1\\2\\1\end{bmatrix}.\nonumber \end{align} $$\int_0^\infty e^{-x^2} dx $$ \begin{equation} [a, b, c]=\left| \begin{array}{ccc} a_1& a_2 & a_3 \\ b_1 &b_2 & b_3 \\ c_1 &c_2 &c_3 \end{array} \right| \end{equation}