Skill: Select an efficient row or column for cofactor expansion

Before expanding, scan the matrix for a row or column with many zeros. Every zero entry contributes a zero term, so choosing that row or column can reduce the amount of work. You may expand along any row or column, as long as you use the correct cofactor signs.

For example, find

$$\det\begin{pmatrix} 1&2&0\\ 0&0&3\\ 4&5&6 \end{pmatrix}.$$

The second row has two zeros, so expand along it. The signs across this row are $-,+,-$:

$$\det\begin{pmatrix} 1&2&0\\ 0&0&3\\ 4&5&6 \end{pmatrix} =0+0-3\det\begin{pmatrix}1&2\\4&5\end{pmatrix}.$$

The $2\times2$ determinant is $1\cdot5-2\cdot4=-3$, so

$$-3(-3)=9.$$

Thus, the determinant is $\boxed{9}$. Choosing the second row made the calculation quicker because its two zero entries required no work.