Suppose, for example, we look at the next to last elementary product in the example above and tabulate the positions of its component numbers.
Entry 3 5
2 Row position 1 2 3 4 Column position 3 4 2 1
Transform the list of column positions by repeatedly swapping pairs of them until you match the list of row positions, and count the number of swaps you need.
Entry 3 5
2 Row position 1 2 3 4 Column position 3 4 2 1 1. 3 ↔ 1 produces 1 4 2 3 2. 4 ↔ 2 produces 1 2 4 3 3. 3 ↔ 4 produces 1 2 3 4 Total number of swaps: 3
The sign of that elementary product is + if the number of swaps is even,
or – if the number of swaps is odd. (The actual number of swaps you
do or the order in which you do them turns out not to matter; for any given
elementary product, you will either always get an even number of swaps, or
always get an odd number of swaps.) In this case, the number of swaps (3)
is odd, so the sign attached to the elementary product is –.
The signed elementary product is then
–(–180) = 180.
In the interactive example above, the elementary products in pink have a + sign attached, and those in blue have a – sign attached.
Given a square matrix A, the determinant of A is the number found by
The determinant of A is denoted by det(A), or by replacing the brackets of A by straight lines: for example, if
The Formal Definition of a Determinant
|Introduction||The formal definition of a determinant||Determinants of special matrices||Using row and column operations on determinants|