Inversion of all principal submatrices of a matrix

Let A_m be an m×m principal submatrix of an infinite-dimensional matrix A. We give a simple formula which expresses A_m+1^-1 in terms of A_m^-1, and based on this formula, an algorithm which computes the inverses of A_m for m=1, 2, 3, ..., n using only 2n³-2n² +n arithmetic operations. This is an improvement over the naive method of computing the inverses separately which would require Σ _m=1ⁿ m³=O(n⁴) arithmetic operations