How do we know the ith of an invertible matrix B is orthogonal to the jth column of B^-1 , if i is not equal/unequal to j?

Question

MathMate · Accepted Answer

First please confirm the following typographical correction indicated in bold:
"How do we know the ith row of an invertible matrix B is orthogonal to the jth column of B^-1 , if i is not equal/unequal to j?"

By definition,
BB^-1=I
which by matrix multiplication, the inner product of the ith row of B and the jth column of B^-1 results in the element Iij, or the element on the ith row and the jth column.

Since the unit matrix I has the property that Iii=1 and Iij=0 if i≠j, the question is answered, namely the ith row of B and the jth column of B^-1 are orthogonal.