a rectangular array of (m X n) elements (numbers) arranged in m columns and n rows:

The Krakówian, introduced in the 1920’s by the Polish astronomer T. Banachiewicz (1882–1954), differs from ordinary matrices used in linear algebra by a simpler multiplication rule. We multiply the columns of cofactors having the same number of rows:

Here rb is the transposed krakowian of b. Furthermore, operations such as division and extraction of a square root, which are lacking in matrix algebra, can be performed on Krakowians. Krakowians have found application in geodetic calculations.


Modrinskii, N. I. Primenenie krakovianov v geodezicheskikh vychisleniiakh. Moscow, 1959.
Zlatanov, G. Krakovianovo smiatane. Sofia, 1968.
Banachiewicz, T. Rachunek krakowianowy z zastosowaniami. Warsaw, 1959.


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Artur Krakowian, 34, said he was left out of pocket after the last incident and he feared the worst again.
The crib is in fact Krakowian in style, but it has Huddersfield features: Castle Hill and a mill chimney, the lion from the Lion Building and two angels - one black, one white.