[Math] Prove that a determinant is zero without using expansion

Question

How to prove that the following determinant is zero without using expansion?

$$\left|\begin{array}{ccc} 1 & 4 & 1 \\
2 & -1 & 0 \\
0 & 18 & 4\end{array}\right|=0$$

I can't get any 2 rows or any 2 columns to be equal and I can't get an entire row or entire column to be zero? What series of operations are required? Thanks.

Edit

I noticed that it is easy to get the diagonal to be zero. I am not sure when one could say if the diagonal is zero then the determinant is zero…

Answer 1

These will do the job for us: $$R_3 \to R_3 - 4R_1$$ $$R_3 \to R_3 + 2R_2$$