<s>[INST] Invertible Matrix Let $A$ be the matrix 
$$ A=\left(\begin{array}{cccc}1&0&1&2\\2&3&\beta&4\\4&0&-\beta&-8\\ \beta&0&\beta&\beta \end{array}\right). $$
For what values of $\beta$ is the matrix invertible? 
 [/INST] Compute the determinat of $A$ with laplace expansion  expanding at second column it will ease the computations because there are there many zeros
$|A|=(-1)^{2+2}3 \left|\begin{array}{ccc}1&1&2\\4&-\beta&-8\\ \beta&\beta&\beta \end{array}\right|$
For $\beta=0 \Rightarrow |A|=0$
Now for $\beta \not =0$
$$|A| \not= 0 \Leftrightarrow \left|\begin{array}{ccc}1&1&2\\4&-\beta&-8\\ 1&1&1 \end{array}\right| \not =0$$
$$\left|\begin{array}{ccc}1&1&2\\4&-\beta&-8\\ 1&1&1 \end{array}\right|=\left|\begin{array}{ccc}1&1&2\\4&-\beta&-8\\ 0&0&-1 \end{array}\right|=\left|\begin{array}{ccc}1&1&2\\0&-\beta-4&-16\\ 0&0&-1 \end{array}\right|=\beta +4$$
Therefore for $\beta \not = 4,0$  $A$ is invertible.
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