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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