Question

Escalone a matriz [0 -1 2 4... 1 2 -2 -3... 3 -1 3 4] .

Answer 1

$\left[\begin{array}{cccc}0&-1&~~2&~~4\\1&~~2&-2&-3\\3&-1&~~3&~~4\end{array}\right]~l_{1}\leftrightarrow l_{2}\\\\\\\left[\begin{array}{cccc}1&~~2&-2&-3\\0&-1&~~2&~~4\\3&-1&~~3&~~4\end{array}\right]~l_{3}\leftarrow l_{3}-3l_{1}\\\\\\\left[\begin{array}{cccc}1&~~2&-2&-3\\0&-1&~~2&~~4\\0&-7&~~9&~~13\end{array}\right]~l_{3}\leftarrow l_{3}-7l_{2}\\\\\\\left[\begin{array}{cccc}1&~~2&-2&-3\\0&-1&~~2&~~4\\0&~~0&-5&-15\end{array}\right]~l_{3}\leftarrow(-\frac{1}{5})l_{3}$

Então, a forma escalonada da matriz é

$\left[\begin{array}{cccc}1&~~2&-2&-3\\0&-1&~~2&~~4\\0&~~0&~~1&~~3\end{array}\right]$

Podemos parar por aí, ou encontrar a forma totalmente escalonada:

$\left[\begin{array}{cccc}1&~~2&-2&-3\\0&-1&~~2&~~4\\0&~~0&~~1&~~3\end{array}\right]~l_{2}\leftarrow(-1)l_{2}\\\\\\\left[\begin{array}{cccc}1&~~2&-2&-3\\0&~~1&-2&-4\\0&~~0&~~1&~~3\end{array}\right]~l_{1}\leftarrow l_{1}+2l_{3},~l_{2}\leftarrow l_{2}+2l_{3}\\\\\\\left[\begin{array}{cccc}1&2&0&3\\0&1&0&2\\0&0&1&3\end{array}\right]~l_{1}\leftarrow l_{1}-2l_{2}\\\\\\\left[\begin{array}{cccc}1&0&0&-1\\0&1&0&~~2\\0&0&1&~~3\end{array}\right]$