Let A and B be two matrices such that the order of A is `5xx7`. If `A^(T)B` and `BA^(T)` are both defined, then (where `A^(T)` is the transpose of matrix A)

Options

A. order of `B^(T)` is `5xx7`
B. order of `B^(T)A` is `7xx7`
C. order of `B^(T)A` is `5xx5`
D. `B^(T)A` is undefined

Answer

To solve the problem, we need to analyze the orders of the matrices involved and the conditions for matrix multiplication to be defined.

### Step-by-Step Solution:

1. **Identify the Order of Matrix A:**
- Given that the order of matrix A is \(5 \times 7\), it has 5 rows and 7 columns.

2. **Determine the Order of A Transpose (A^T):**
- The transpose of matrix A, denoted as \(A^T\), will have its rows and columns swapped. Thus, the order of \(A^T\) is \(7 \times 5\).

3. **Analyze the Expression \(A^T B\):**
- For the product \(A^T B\) to be defined, the number of columns in \(A^T\) must equal the number of rows in matrix B.
- Since \(A^T\) has 5 rows, matrix B must have 5 rows. Therefore, the number of columns in B must be 7 (to match the number of columns in \(A^T\)).
- Thus, the order of matrix B is \(5 \times 7\).

4. **Analyze the Expression \(BA^T\):**
- For the product \(BA^T\) to be defined, the number of columns in B must equal the number of rows in \(A^T\).
- Since \(A^T\) has 7 columns, matrix B must have 7 columns. We already established that B has 5 rows, so the order of B is indeed \(5 \times 7\).

5. **Determine the Order of \(B^T\):**
- The transpose of matrix B, denoted as \(B^T\), will have its rows and columns swapped. Thus, the order of \(B^T\) is \(7 \times 5\).

6. **Evaluate the Options:**
- **Option 1:** The order of \(B^T\) is \(5 \times 7\) (Incorrect, as it is \(7 \times 5\)).
- **Option 2:** The order of \(B^T A\) is \(7 \times 7\) (Correct, as \(B^T\) has 7 rows and A has 7 columns).
- **Option 3:** The order of \(B^T A\) is \(5 \times 5\) (Incorrect, as it is \(7 \times 7\)).
- **Option 4:** \(B^T A\) is undefined (Incorrect, as it is defined).

### Conclusion:
The correct option is **Option 2**: The order of \(B^T A\) is \(7 \times 7\).