Answer
the vectors of $S$ are not linearly independent.
Work Step by Step
The set
$S=\left\{\left[\begin{array}{cc}{1} & {0} \\ {0} & {0}\end{array}\right],\left[\begin{array}{cc}{0} & {1} \\ {1} & {0}\end{array}\right],\left[\begin{array}{cc}{1} & {0} \\ {0} & {1}\end{array}\right],\left[\begin{array}{rr}{8} & {-4} \\ {-4} & {3}\end{array}\right]\right\}$
is not a basis for $M_{2,2}$ because the vectors of $S$ are not linearly independent. For example, we have
$$\left[\begin{array}{rr}{8} & {-4} \\ {-4} & {3}\end{array}\right]=5\left[\begin{array}{cc}{1} & {0} \\ {0} & {0}\end{array}\right]-4\left[\begin{array}{cc}{0} & {1} \\ {1} & {0}\end{array}\right]+3\left[\begin{array}{cc}{1} & {0} \\ {0} & {1}\end{array}\right].$$
Which means that $S$ is a set of linearly dependent vectors.
