Answer
$ a\neq 0, b=c=d= 0$.
Work Step by Step
We have the matrix
$$ \left[ \begin {array}{ccc} a&b\\ c&d
\end {array} \right].
$$
Multiply the first row by $c$ and adding it to $-a$ times the second row, we get
$$\left[ \begin {array}{ccc} a&b\\ 0&bc-ad\end {array} \right].
$$
Adding $-b$ times the second row to $bc-ad$ times the first row, we get
$$\left[ \begin {array}{ccc} a(bc-ad)&0\\ 0&bc-ad\end {array} \right].
$$
Now, the matrix $ \left[ \begin {array}{ccc} a&b\\ c&d
\end {array} \right]$ is row-equivalent to $ \left[ \begin {array}{ccc} 1&0\\ 0&1
\end {array} \right]$ if $bc-ad=0$ and this impossible because it turns the matrix to the zero matrix. But we have to restrict the choice of $a,b,c,d$ from the begging, that is $ a\neq 0, b=c=d= 0$.
