nth Order Determinant: Permutation Definition and Term-by-Term Expansion
Term-by-Term Calculation Process
| # | Permutation σ | Inversion Count τ | Sign | Product Term | Value | Signed Term | Cumulative Sum |
|---|
Basic Introduction
The permutation definition of an nth order determinant is:
det(A)=∑((-1)^τ(σ)·a1σ1·a2σ2·...·anσn), summing over all n! permutations.
Here τ(σ) is the number of inversions in permutation σ. The inversion count determines whether the term is positive or negative.
Learning Tips
- For n=2, 3: calculate manually first and observe the sign pattern.
- For n=4, 5: observe the factorial growth of terms, understand "clear definition but computationally expensive".
- Compare with Gaussian elimination results to verify correctness.