The determinant of a square matrix is built out
of "signed elementary products" of the numbers in that matrix.

Each elementary product has an associated sign which depends on the rows
and columns its numbers come from. Here's how to find that sign.

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 5–4

2Row position 1 2 3 4Column 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 5–4

2Row position 1 2 3 4Column position 3 4 2 11. 3 ↔ 1 produces 1 4 2 32. 4 ↔ 2 produces 1 2 4 33. 3 ↔ 4 produces 1 2 3 4Total 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.

Once we have the signed elementary products, we
can define a determinant.

Given a square matrix A, the determinant of A is the number found by

- calculating all elementary products of A
- attaching the appropriate sign to each elementary product
- adding the results.

The determinant of A is denoted by * det*(A),
or by replacing the brackets of A by straight
lines: for example, if

then

.

Needless to say, calculating a determinant by
using the formal definition would consume a horrendous amount of time, even
for quite small matrices. In practice, we don't use the definition to calculate
determinants, but we do use it to develop rules that help us calculate determinants
more efficiently.

The Formal Definition of a Determinant |
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Introduction | The formal definition of a determinant | Determinants of special matrices | Using row and column operations on determinants |