To give some context, recently in one of our classes, our professor introduced solving the determinant of arbitrary-sized matrices. On one of his examples, he asked us to calculate the matrix $$\begin{bmatrix} 1 & 2 & 3 & 4 \\ 5 & 6 & 7 & 8 \\ 9 & 0 & 1 & 2 \\ 3 & 4 & 5 & 6 \\ \end{bmatrix}$$ Calculating this is tedious, however, one of the students raised his hand shortly after, and answered 0, which was correct.
What I want to know is how he arrived at that solution so quickly, I want to fill the gap in my knowledge.
Subtracting the fourth row from the second gives $(2,2,2,2)$; adding twice this to the first row gives the second row. Thus: $$R_1+2(R_2-R_4)=0$$ and the matrix has linearly dependent rows, hence determinant zero.
These operations are very easy to spot, given that the first, second and fourth rows contain consecutive integers.