Ok, actually its actually trivial now that I think about it.

We want matrix T and matrix M with transpose of M being M^{T} such that

TM=M^{T}

This is obviously satisfied if T=M^{T}M^{-1} since

TM=M^{T}M^{-1}M=M^{T}

This requires that M is invertible, which is the determinant requirement I mentioned above.

I think you never see this equation anywhere because computers are far more efficient at just doing the transpose than they are at matrix multiplication.