If A and B are invertible matrices of the same order, then `(AB)'` is equal to

Options

A. `A'B'`
B. `B'A'`
C. `-A'B'`
D. `-B'A'`

Answer

To find the transpose of the product of two invertible matrices \( A \) and \( B \), we can use the property of transposes of matrices.

### Step-by-Step Solution:

1. **Understanding the Problem**:
We need to find the transpose of the product of two matrices \( AB \), where \( A \) and \( B \) are invertible matrices of the same order.

2. **Using the Transpose Property**:
The property of transposes states that for any two matrices \( A \) and \( B \):
\[ (AB)^T = B^T A^T \]
This means that the transpose of the product of two matrices is equal to the product of their transposes in reverse order.

3. **Applying the Property**:
Since \( A \) and \( B \) are both invertible, we can apply the property:
\[ (AB)^T = B^T A^T \]

4. **Conclusion**:
Therefore, the transpose of the product \( AB \) is given by:
\[ (AB)^T = B^T A^T \]

### Final Answer:
\[ (AB)^T = B^T A^T \]