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[itex]A^{j}_{i}[/itex]*[itex]A_{j}^{k}[/itex]=[itex]\delta_{i}^{k}[/itex]

It is true because it is the result of a calculation. But assuming it is true ¿What happens if one of the [itex]A_{i}^{j}[/itex] is zero?.

Then it does not matter wich is the value of the [itex]A_{j}^{k}[/itex] the equality will be false because 0*any number= 0, but supposedly it should be 1, due to delta having i=k.

So the point is ¿What happens if the coefficients are zero?¿Is it really true?.