2
votes
2answers
46 views

Proof that $\det(A)=\det(A^T)$ using permutations.

I'm reading a proof for the identity $\det(A) = \det(A^T)$ and I'm trying to udnerstand why the following rows are equivalent: $$\eqalign{ & \det ({A^T}) = \sum\limits_{\pi \in {S_n}} ...
0
votes
1answer
53 views

How to prove that determinant with permutation symbols

How to prove that $$\varepsilon_{ijk}a_{i\ell}a_{jm}a_{kn} = \det[a]\epsilon_{\ell mn}$$ I'm trying to solve this problem with permutation symbol but i can't solve it Help me,please. Thank you ...
5
votes
1answer
57 views

possible determinants of permutations

this is taken from Gilbert Strang's Linear Algebra book: What are all the possible $4\times4$ determinants of $I + P_{even}$? (P - permutation matrix) I seem to be stuck on this question except for ...
3
votes
3answers
155 views

Is there a way to get all the permutations of $S_4$

I need to calculate the determinant of a $4 \times 4$ matrix by "direct computation", so I thought that means using the formula $$\sum_{\sigma \in S_4} (-1)^{\sigma}a_{1\sigma(1)}\ldots ...
2
votes
2answers
96 views

Bijection from $S_{n-1}$ to $\{\sigma \in S_{n} : \sigma(k) = j \}$

Let $n$ be a natural number. Let $k$ be an element of $\{1, \ldots , n\}$. For each j in $\{1, \ldots , n\}$, I want to find a bijection $f_j$ from $S_{n-1}$ to $\{\sigma \in S_n : \sigma(k) = j ...
15
votes
2answers
230 views

Help deriving that $\mathrm{sign} : S_n\to \{\pm 1\}$ is multiplicative

$\def\sign{\operatorname{sign}}$ For homework, I am trying to show that $\sign:S_n \to \{\pm 1\}$ is multiplicative, i.e. that for any permutations $\sigma_1,\sigma_2$ we have $$\sign(\sigma_1 ...