Determinants

Determinant function

A $n \times n$ determinant is a function

det : ℳ_{n \times n} \to 𝕂

such that

det (E A) = det (E) det (A)

for any elementary matrix $E$ and any matrix $A$ . Furthermore

$det (E) = 1$ if $E$ represents a linear combination;
$det (E) = -1$ if $E$ represents a swap;
$det (E) = k$ if $E$ represents a rescaling by $k$ ;
$det (I) = 1$

remark

Conditions (1) and (3) imply condition (2).

row or zeros

If matrix has a row of zeroes then its determinant is zero.

linearly dependent rows

The determinant of a matrix is zero if and only if its rows are linearly dependent.

uniqueness

The determinant function is unique.

multiplication

det (A B) = det (A) det (B)

transpose

det (A) = det (A^{t})

Existence

Permutation

(\begin{matrix} 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{matrix})

Sum of vectors

det (\begin{matrix} {\vec{ρ}}_{1} \\ ⋮ \\ {\vec{ρ}}_{i - 1} \\ \vec{𝒖} + \vec{𝒗} \\ {\vec{ρ}}_{i + 1} \\ ⋮ \\ {\vec{ρ}}_{n} \end{matrix}) = det (\begin{matrix} {\vec{ρ}}_{1} \\ ⋮ \\ {\vec{ρ}}_{i - 1} \\ \vec{𝒖} \\ {\vec{ρ}}_{i + 1} \\ ⋮ \\ {\vec{ρ}}_{n} \end{matrix}) + det (\begin{matrix} {\vec{ρ}}_{1} \\ ⋮ \\ {\vec{ρ}}_{i - 1} \\ \vec{𝒗} \\ {\vec{ρ}}_{i + 1} \\ ⋮ \\ {\vec{ρ}}_{n} \end{matrix})

Sum over permutations

\begin{matrix} det (A) & = & \sum_{𝝈} a_{1 𝝈 (1)} a_{2 𝝈 (2)} \dots a_{n 𝝈 (n)} det (P_{𝝈}) \\ = & \sum_{i = 1}^{n} {(-1)}^{i + c} a_{i c} det (A (i ∣ c)) \\ = & \sum_{i = 1}^{n} {(-1)}^{r + i} a_{r i} det (A (r ∣ i)) \end{matrix}

Determinant of Permutations

Inversions

Let $𝝈 = (𝝈 (1), 𝝈 (2) \dots 𝝈 (n))$ be a permutation. In the permutation matrix $P_{𝝈}$ two rows $k < l$ are inversion if and only if $𝝈 (k) > 𝝈 (l)$

Swaps

A row swap in a permutation matrix changes the parity of the number of inversions in the matrix.

Preservations of parity

If a permutation has odd number of inversions then swapping to the identity matrix takes odd number of inversions. If a permutation has even number of inversions then swapping the rows to the indetity matrix takes odd number of inversions.

Sign of a permutation

The sign of a permutation $𝝈 = (𝝈 (1), 𝝈 (2) \dots 𝝈 (n))$ is

sign (𝝈) = {(-1)}^{# inversion 𝝈}

Determinant formulae

\begin{matrix} det (A) & = & \sum_{𝝈} a_{1 𝝈 (1)} a_{2 𝝈 (2)} \dots a_{n 𝝈 (n)} det (P_{𝝈}) \\ = & \sum_{𝝈} a_{1 𝝈 (1)} a_{2 𝝈 (2)} \dots a_{n 𝝈 (n)} sign (P_{𝝈}) \end{matrix}