How to determine if a set of five $2\times2$ matrices is independent

Since the space of all $2\times2$ matrices is $4$-dimensional, every set of $5$ such matrices is linearly dependent.

As has been pointed out, four matrices form a basis for the $2\times2$ matrices (the easiest would be $$ \left[\begin{matrix}1&0\\0&0\end{matrix}\right], \left[\begin{matrix}0&1\\0&0\end{matrix}\right], \left[\begin{matrix}0&0\\1&0\end{matrix}\right], \left[\begin{matrix}0&0\\0&1\end{matrix}\right] $$) so five matrices cannot be linearly dependent.

In your case the dependence is $$ \left[\begin{matrix}1&2\\2&1\end{matrix}\right] + \left[\begin{matrix}2&1\\-1&2\end{matrix}\right] + \left[\begin{matrix}0&1\\1&2\end{matrix}\right] - 2\left[\begin{matrix}1&0\\1&1\end{matrix}\right] - \left[\begin{matrix}1&4\\0&3\end{matrix}\right] = \left[\begin{matrix}0&0\\0&0\end{matrix}\right]. $$

As the others have said, this set of $5$ must be linearly dependent because the dimension of the space of all $2\times 2$ matrices is $4$.

More generally, how do you show that a set of vectors is linearly dependent or independent? Create a linear combination of the vectors, set it equal to $0$, and try to solve it.

$$ a_1X_1 + a_2X_2 + \dotsb+ a_nX_n = 0 $$ If the only possible solution is $a_1 = a_2 = \dotsb = a_n = 0$ then the set is independent. If a different solution exists then the set is dependent.