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.