Proving a property of transposition

Question

I'm studying for a Linear exam and I can't seem to figure out how to prove the following:

$$(λA)^T = \lambda.A^T$$

I'm sure the proof is simple I just can't get it. It should be done without using determinants of any sort.

Answer 1

Writing $B = (B_{i,j})$ for any matrix $B$, $$[(\lambda A)^T]_{i, j} = (\lambda A)_{j,i} = \lambda A_{j,i}$$ and $$[\lambda A^T]_{i,j} = \lambda (A^T)_{i,j} = \lambda A_{j,i}.$$