We are given that, 

Order of matrix A = 2 × 3 

A^TB and BA^T are defined matrices. 

We need to find the order of matrix B. 

We know that the transpose of a matrix is a new matrix whose rows are the columns of the original. 

So, 

If the number of rows in matrix A = 2 

And, number of columns in matrix A = 3 

Then, 

The number of rows in matrix A^T = number of columns in matrix A = 3 Number of columns in matrix A^T = number of rows in matrix A = 2 

So, 

Order of matrix A^T can be written as,

Order of matrix A^T = 3 × 2 

Thus, 

We have Number of rows of A^T = 3 …(i) 

Number of columns of A^T = 2 …(ii) 

If A^TB is defined, that is, it exists, then 

Number of columns in A^T = Number of rows in B 

⇒ 2 = Number of rows in B [from (ii)] 

Or,

Number of rows in B = 2 …(iii) 

If BA^T is defined, that is, it exists, then

Number of columns in B = Number of rows in A^T 

Substituting value of number of rows in A^T from (i), 

⇒ Number of columns in B = 3 …(iv) 

From (iii) and (iv), 

Order of B = Number of rows × Number of columns 

⇒ Order of B = 2 × 3 

Thus, 

Order of B is 2 × 3