Matrix A has order m×n (m rows, n columns).
Matrix B exists such that ABT and BTA are both well-defined.
For matrix multiplication to be possible, the number of columns in the first matrix must equal the number of rows in the second matrix.
Let matrix B have order p×q (p rows, q columns).
Then BT (transpose of B) will have order q×p.
For ABT to be well-defined:
A is (m×n)
BT is (q×p)
Therefore, n=q
For BTA to be well-defined:
BT is (q×p)
A is (m×n)
Therefore, p=m
From the two conditions:
p=m
q=n
Since B has order p×q:
Order of matrix B =m×n