In astronomical and geodetic calculations, Cracovians are a clerical convenience introduced in 1925 by Tadeusz Banachiewicz for solving systems of linear equations by hand.
The Cracovian product of two matrices, say A and B, is defined by A ∧ B = BTA, where BT and A are assumed compatible for the common (Cayley) type of matrix multiplication.
Since (AB)T = BTAT, the products (A ∧ B) ∧ C and A ∧ (B ∧ C) will generally be different; thus, Cracovian multiplication is non-associative.
Use of Cracovians in astronomy faded as computers with bigger random access memory came into general use.
Specifically, the Cracovian product of matrices A and B can be obtained as crossprod(B, A).