Why are there minuses in calculating determinants?

The two-by-two determinant

eq0028P

Why minus?

Here’s an explanation. See also: Why is the rule as it is for inverting 2×2 matrices?.

If the determinant measures how big the area of the image of a unit area is, then the determinant is half the area of the triangle OAB in this diagram, since OAB is the image of the triangle with corners at (0,0), (1,0), and (0,1), which has area ½

area-determinant

The area of OAB depends partly on how big a, b, c, and d are, but also partly on how wide the angle is between OA and OB. If O, A, and B are exactly in line, then the area of the image is zero however big a, b, c, and d are.

The slope of OA is c/a

The slope of OB is d/b

The difference in the slopes, which is a measure of the angle between them, is d/b−c/a

But d/b−c/a is (ad−bc)/ab

There it is – ad−bc! The minus comes from the fact that the determinant has to include some measure of how much the transformation represented by the matrix leaves the axes skew to each other, and how much it narrows down the area by bringing them closer to each other.

In general an n×n determinant A is calculated as the sum over all i1, i2, i3…., in of

εi1i2i3….inA1i1A2i2…Anin

where εi1i2i3….in is defined to equal 0 if i1, i2, i3…., in is not a permutation of 1, 2, 3…. n (i.e. is not just those same numbers 1, 2, 3…. n but in a different order)

and to equal 1 if i1, i2, i3…., in is an even permutation of 1, 2, 3…. n

and to equal −1 if i1, i2, i3…., in is an odd permutation of 1, 2, 3…. n

(Click here for an explanation of what even and odd permutations are).

In other words, it is the total of all the n! products you can get by choosing n elements, all in different rows and all in different columns, and multiplying them together, with a minus sign on each product where there is an odd permutation.

The minus signs are there so that the determinant can include some measure of how much the transformation represented by the matrix leaves the n axes skew to each other, and how much it narrows down the volume or hypervolume by bringing them closer to each other.

Looking at it another way, the minus signs are there for exactly the same reason as there are minus signs in the working-out of the vector cross-product (a1i+a2j+a3k)×(b1i+b2j+b3k), which also measures an area.