Here is the theorem we need to prove.

Theorem. The following properties hold:

  1. (AT)T=A, that is the transpose of the transpose of A is A (the operation of taking the transpose is an involution).
  2. (A+B)T=AT+BT, the transpose of a sum is the sum of transposes.
  3. (kA)T=kAT.
  4. (AB)T=BTAT, the transpose of a product is the product of the transposes in the reverse order.

Proof. 1. The (i,j)-entry of AT is the (j,i)-entry of A, so the (i,j)-entry of (AT)T is the (j,i)-entry of AT, which is the (i,j)-entry of A. Thus all entries of (AT)T coincide with the corresponding entries of A, so these two matrices are equal.

2. The (i,j)-entry of AT+BT is the sum of (i,j)-entries of AT and BT, which are (j,i)-entries of A and B, respectively. Thus the (i,j)-entry of AT+BT is the (j,i)-entry of the sum of A and B, which is equal to the (i,j)-entry of the transpose (A+B)T.

3. The proof is similar.

4. Compare the (i,j)-entries of (AB)T and BTAT.