Better I think would be to say "the result in column i and row j is the sum of product of elements in column i of the left cracovian and column j of the right cracovian".

And even by this definition the example given doesn't seem to track (and the strangeness of sometimes saying "+" and sometimes not, and having both "0" and "-0" in the example is bananas!):

   {  3  2 } {  1  -4 }  =   {  5   -2 }
   { -1  0 } { -2   3 }  =   {  0    2 }


   3 * 1 + -1 * -2 == 5 -- check
   3 * -4 + -1 * 3 == -15 -- what?
   2 * 1 + 0 * -2 == 2 (okay, but shouldn't this be in the lower left, column 1 dotted with column 2?)
   2 * -4 + 0 * 3 = -8 (now I'm really missing something)

Old texts got really worked up whether a vector was a row or column. The programming language APL resolved this quite nicely: A scalar has no dimensions, a vector has its length as its one dimension, ... Arbitrary rank objects all played nicely with each other, in this system.

A Cracovian is a character or two's difference in APL code. There's a benign form of mental illness learning anything, where one clutches onto something novel and obsesses over it, rather than asking "That was exciting! What novel idea will I learn in the next five minutes?" I have friends from my working class high school who still say "ASSUME makes an ask of you and me" as if they just heard it for the first time, while the most successful mathematicians that I know keep moving like sharks.

I wouldn't stall too long thinking about Cracovians, as amusing a skim as the post provided.