To derive the determinant of a square matrix, we can always use elimination to convert it into an upper triangular matrix. For example, in the $M_{2 \times 2}$ case, we can always use elimination to convert the $M_{2 \times 2}$ to an upper triangular matrix. There are two cases to consider: when $a \neq 0$ and when $a=0$.
I understand the case where $a \neq 0$.
If $a \neq 0$, then
$\begin{bmatrix}a&b\\c&d\end{bmatrix} \rightarrow \begin{bmatrix}a&b\\0&d-\frac{c}{a}b\end{bmatrix}$
We can then multiply the pivots along the diagonal line (i.e. $a \dot (d-\frac{c}{a}b) = ad - bc$)
But I am not sure the case when $a = 0$ (or one of the diagonal entries is $0$). Do I need to exchange rows? What properties of determinant do I need to use? I know the determinant is $0$ when the matrix is singular.
Edit: I am so sorry if the comment causes any confusion. It was due to a typo in the title. Initially, the title was: Determinant when one of the diagonal entries of a diagonal matrix is $0$. But, I actually meant to ask for the Determinant of a triangular matrix when one of its diagonal entries is $0$. This means that the counterexample $\begin{bmatrix}1&1\\1&0\end{bmatrix}$ no longer works. The second last comment clears my concern :)