Product of Determinants

Two determinants of the same order can be multiplied to yield a third determinant of that order, using any one of four equivalent schemes:

1. row-by-column (the ordinary matrix-multiplication rule),
2. row-by-row,
3. column-by-column, or
4. column-by-row.

All four produce the same value. This works because $|A^T|=|A|$ (Property 1) — treating a determinant "by its rows" or "by its columns" makes no difference to its value, so any of the four pairings is legitimate.

The underlying algebraic fact is simply $|AB|=|A|\,|B|$ (the product rule). And although matrix multiplication itself is not commutative ($AB\ne BA$ in general), the corresponding determinants always agree: $|AB|=|BA|$, because both equal the product $|A|\,|B|=|B|\,|A|$ of two ordinary numbers. …