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