Give a combinatorial interpretation for calculating the inverse of a matrix.

Basically, the formula for {$A^{-1}_{ij}$} has us look at all permutations that have edge {$a_{ji}$} and set that edge equal to 1. And it adds a sign of {$(-1)^{j-i}$}. Thus the inverse will list permutational paths from i to j. When we multiply the original matrix with its inverse, then we will be creating combinations of an edge {$a_{ij}$} and a signed permutation path from i to j. When i=j we should get all permutations, which yields the determinant, and so dividing by that we get 1. And when i and j are different, than we should get all terms paired with opposite signs, yielding 0. And what does that all mean?

**Readings**

- Invertible matrix see Analytic method
- Minor see Cofactor

Parsiųstas iš http://www.ms.lt/sodas/Book/InverseMatrixFormula

Puslapis paskutinį kartą pakeistas 2019 birželio 17 d., 16:06