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 determine the order of matrix B and the order of the product \( B^T A \) given that the order of matrix A is \( 5 \times 7 \).

### Step-by-Step Solution:

1. **Identify the Order of Matrix A**:
The order of matrix A is given as \( 5 \times 7 \). This means A has 5 rows and 7 columns.

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

3. **Condition for \( A^T B \) to be Defined**:
For the product \( A^T B \) to be defined, the number of columns in \( A^T \) must equal the number of rows in B. Since \( A^T \) has 5 rows, we denote the order of matrix B as \( p \times q \). Thus, we have:
\[ p = 5 \]
So, the order of B can be expressed as \( 5 \times q \).

4. **Condition for \( B A^T \) to be Defined**:
For the product \( B A^T \) to be defined, the number of columns in B must equal the number of rows in \( A^T \). Since \( A^T \) has 5 columns, we have:
\[ q = 7 \]
Therefore, the order of B is \( 5 \times 7 \).

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 \( B^T \) will be \( 7 \times 5 \).

6. **Determine the Order of the Product \( B^T A \)**:
Now we need to find the order of the product \( B^T A \). The order of \( B^T \) is \( 7 \times 5 \) and the order of A is \( 5 \times 7 \). For the product \( B^T A \) to be defined, the number of columns in \( B^T \) (which is 5) must equal the number of rows in A (which is also 5). The resulting order of the product will be given by the outer dimensions:
\[ \text{Order of } B^T A = 7 \times 7 \]

### Final Result:
- The order of \( B^T \) is \( 7 \times 5 \).
- The order of \( B^T A \) is \( 7 \times 7 \).