math111_logo Proof of Properties

1. Transpose

We prove that the determinant is not changed by the transpose.

If we denote the ij-entry of A by aij, then the ij-entry of AT is aji. Consequently, if we change aij to aji in the explicit formula

detA = ∑all permutations sign(i1, i2, ..., in) ai11ai22...ainn,

then we get

detAT = ∑all permutations sign(i1, i2, ..., in) a1i1a2i2...anin.

To see why detAT can be equal to detA, let us consider the 4 by 4 case. Consider a term sign(3, 1, 4, 2)a13a21a34a42 in detAT. By looking at the product of entries, we have

a13a21a34a42 = a21a42a13a34.

Correspondingly, we have the term sign(2, 4, 1, 3)a21a42a13a34 in detA. It turns out that sign(3, 1, 4, 2) = sign(2, 4, 1, 3), so that the corresponding terms in detAT and detA are equal (together with the sign).

In general, a similar argument can be carried out as follows. Any term sign(i1, i2, ..., in) a1i1a2i2...anin in detAT can be rewritten as sign(i1, i2, ..., in) aj11aj22...ajnn, which is compared with the corresponding term sign(j1, j2, ..., jn) aj11aj22...ajnn in detA. The equality detAT = detA will then be the consequence of the equality sign(j1, j2, ..., jn) = sign(i1, i2, ..., in).

To prove sign(j1, j2, ..., jn) = sign(i1, i2, ..., in), we first need to understand the rewriting process. In more detail, the process is the following. We start with a 2 by n matrix

first indices in detAT [ 1 2 ... n ].
second indices in detAT i1 i2 ... in

A sequence of transpositions that changes (i1, i2, ..., in) to (1, 2, ..., n) can be applied to the columns of the matrix, giving us a sequence of transpositions on 2 by n matrices

[ 1 2 ... n ] → ... → [ j1 j2 ... jn ]. ⇐ first indices in detA
i1 i2 ... in 1 2 ... n ⇐ second indices in detA

For the 4 by 4 example a13a21a34a42, this means the following

[ 1 2 3 4 ][ 2 1 3 4 ][ 2 1 4 3 ][ 2 4 1 3 ],
3 1 4 2 1 3 4 2 1 3 2 4 1 2 3 4

where we note that the result corresponds to a21a42a13a34. The upshot is that, the sequence of transpositions that changed (3, 1, 4, 2) to (1, 2, 3, 4) also changes (1, 2, 3, 4) to (2, 4, 1, 3). Equivalently, the transpositions in the sequence

(2, 4, 1, 3) → (2, 1, 4, 3) → (2, 1, 3, 4) → (1, 2, 3, 4)

is the reverse of the transpositions in the sequence

(3, 1, 4, 2) → (1, 3, 4, 2) → (1, 3, 2, 4) → (1, 2, 3, 4).

Since the reversing does not change the number of transpositions, we conclude that sign(2, 4, 1, 3) = sign(3, 1, 4, 2).

In general, given a sequence of transpositions

(i1, i2, ..., in) → ... → (1, 2,..., n),

the reverse sequence of transpositions gives us

(j1, j2, ..., jn) → ... → (1, 2,..., n).

Since the two sequences consist of the same number of transpositions, we have sign(j1, j2, ..., jn) = sign(i1, i2, ..., in).


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