# LaTeX macro examples

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## Equation

\begin{align*}
x^2 + y^2 &= 1 \\
y &= \sqrt{1 - x^2}
\end{align*}

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## Matrix operation

C=A+BD=A⋅B\\begin{align\*} C &= A + B \\\\ D &= A \\cdot B \\end{align\*}CD​\=A+B\=A⋅B​

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## Left aligned

x2+y2=1y=1−x2\\begin{align\*} x^2 + y^2 &= 1 \\\\ y &= \\sqrt{1 - x^2} \\end{align\*}x2+y2y​\=1\=1−x2​​

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## Inline

The photon traces a longer diagonal path, so the same tick stretches to 

\Delta t = \gamma\,\Delta t_0

 where the Lorentz factor is 

\gamma = (1 - v^2/c^2)^{-1/2}

. Because velocity is always below light speed, 

v/c < 1

 and so 

\gamma \geq 1

 always, growing without bound as 

v \to c

. At everyday speeds 

\gamma \approx 1

and the effect vanishes. Cosmic-ray muons make this concrete. With a rest-frame lifetime of 

\tau_0 \approx 2.2\,\mu s

they should decay long before reaching the ground, yet we detect them at sea level. Travelling at 

v \approx 0.998c

 gives 

\gamma \approx 15.8

, stretching their observed lifetime to 

\Delta t \approx 35\,\mu s

\- just long enough to survive the trip.

The photon traces a longer diagonal path, so the same tick stretches to 

\Delta t = \gamma\,\Delta t_0

 where the Lorentz factor is 

\gamma = (1 - v^2/c^2)^{-1/2}

. Because velocity is always below light speed, 

v/c < 1

 and so 

\gamma \geq 1

 always, growing without bound as 

v \to c

.

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## Large Inline

this is some latex text before

E = mc^2

and afterwards, this is some latex text before

E = mc^2

and afterwards, this is some latex text before

E=mc^2\begin{pmatrix}12 & 12 & 3\\ 5 & 4 & 3\\ f & 3 & e\end{pmatrix}

and afterwards, this is some latex text before

E = mc^2

and afterwards,