The standard approach uses the top row:

And the determinant is:

Where the sign of the second term is negative by the checkerboard rule:

Multiplying out we obtain:

The 6 terms contain three components, each taken from a different row and column. For example, the components of

`bdi`

are from:The checkerboard rule makes the sign come out correctly.

If we're working with the top row or the middle column and so processing

`b x (di - fg)`

or `d x (bi - ch)`

, we'll need the minus sign; whereas if we're obtaining this term from `i x (ae - bd)`

we already have a minus sign.Let's try using the last row. We have:

Compare with the first example to see that all the terms are present.

Can we do it by the diagonal? Try

`ceg`

:Nope. Some terms are correct, but some are duplicates.