Answer
$$a\neq 0, \quad bc-ad\neq 0.$$
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
$$\left[ \begin {array}{ccc} a&b\\ 0&bc-ad\end {array} \right].
$$
Dividing the second row on $bc-ad$, we get
$$\left[ \begin {array}{ccc} a&b\\ 0&1\end {array} \right].
$$
Multiply the second row by $-b$ and adding it to the first row, we get
$$\left[ \begin {array}{ccc} a&0\\ 0&1\end {array} \right].$$
Dividing the first row on $a$, we get
$$ \left[ \begin {array}{ccc} 1&0\\ 0&1
\end {array} \right].$$
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]$ provided that
$$a\neq 0, \quad bc-ad\neq 0.$$
