Answer
$ad-cb\neq 0$
Work Step by Step
The system can be written as follows
$$
\left[\begin{array}{rrr}{a} & {b} \\ {c} & {d}\end{array}\right]\left[\begin{array}{rrr}{x} \\ {y} \end{array}\right]=\left[\begin{array}{rrr}{e} \\ {f} \end{array}\right].
$$
Now, the above system has solution if and only if the determinant of the coefficient matrix has inverse, that is, its determinant is non zero. Moreover, the solution is given by
$$\left[\begin{array}{rrr}{x} \\ {y} \end{array}\right]=\frac{1}{ad-cb}\left[\begin{array}{rrr}{d} & {-b} \\ {-c} & {a}\end{array}\right]\left[\begin{array}{rrr}{e} \\ {f} \end{array}\right].$$
It is easy to see that the above solution is unique and exists if and only if $ad-cb\neq 0$ that is the determinant of the coefficient matrix is nonzero.
