Linear Algebra, 2020a

Section I. Definition 343
Computing a determinant with the permutation expansion typically takes
longer than with Gauss’s Method. However, we will use it to prove that the
determinant function exists. The proof is long so we will just state the result
here and defer the proof to the following subsection.
3.14 Theorem For each n there is an n×n determinant function.
Also in the next subsection is the proof of the next result (they are together
because the two proofs overlap).
3.15 Theorem The determinant of a matrix equals the determinant of its transpose.
Because of this theorem, while we have so far stated determinant results in
terms of rows, all of the results also hold in terms of columns.
3.16 Corollary A matrix with two equal columns is singular. Column swaps
change the sign of a determinant. Determinants are multilinear in their columns.
Proof For the first statement, transposing the matrix results in a matrix with
the same determinant, and with two equal rows, and hence a determinant of
zero. Prove the other two in the same way.
QED
We finish this subsection with a summary: determinant functions exist, are
unique, and we know how to compute them. As for what determinants are
about, perhaps these lines [Kemp] help make it memorable.
Determinant none,
Solution: lots or none.
Determinant some,
Solution: just one.
Exercises
This summarizes our notation for the 2- and 3-permutations.
i 1 2 i 1 2 3
φ 1 (i) 1 2 φ 1 (i) 1 2 3
φ 2 (i) 2 1 φ 2 (i) 1 3 2
φ 3 (i) 2 1 3
φ 4 (i) 2 3 1
φ 5 (i) 3 1 2
φ 6 (i) 3 2 1
̌ 3.17 For this matrix, find the term associated with each 3-permutation.
⎛ ⎞
1 2 3
M = ⎝4 5 6⎠
7 8 9