 The Cracovian product of two matrices, say "A" and "B", is defined by "A" ∧ "B" = "B"T"A", where "B"T and "A" are assumed compatible for the common (Cayley) type of matrix multiplication.Since ("AB")T = "B"T"A"T, the products ("A" ∧ "B") ∧ "C" and "A" ∧ ("B" ∧ "C") will generally be different; thus, Cracovian multiplication is non-associative.Cracovians are an example of a quasigroup.