Answer
$S$ is not linearly independent set of vectors.
Work Step by Step
Consider the combination
$$a \left[\begin{array}{ll}{1} & {0} \\ {0} & {-2}\end{array}\right]+b\left[\begin{array}{cc}{0} & {1} \\ {1} & {0}\end{array}\right]+c\left[\begin{array}{ll}{-2} & {1} \\ {1} & {4}\end{array}\right]=\left[\begin{array}{ll}{0} & {0} \\ {0} & {0}\end{array}\right], \quad a,b,c\in R.$$
Which yields the following system of equations \begin{align*} a-2c&=0\\ b+c&=0\\ -2a+4c&=0. \end{align*} The determinant of the coefficient matrix of the above system is given by $$ \left| \begin {array}{cccc} 1&0& -2\\ 0&1&1 \\-2&0&4\end {array} \right|=0. $$ Since the determinant of the coefficient matrix is zero, then the system has non zero solutions and hence, $S$ is not linearly independent set of vectors.
