Prove the theorem (AB)^T = B^TA^T.

Complete the first step of the proof by filling in the blank.

The (i, j)-entry of (AB)^T is the (j, i)-entry of AB, which is a_j1 b_1i +... + a_jn b_ni.

Complete the second step of the proof by filling in the blank.

The entries in row i of B^T are b _1i, b_ni.

Complete the third step of the proof by filling in the blank.

The entries in column j of A^T are a_j1, a_jn.

Complete the fourth step of the proof by filling in the blank.

The (i, j)-entry in B^TA^T is a_j1 b_1i +... + a_jn b_ni.

Write a conclusion by filling in the blank.

Therefore. (AB)^T = B^TA^T.