Step 1: Understanding the Question:
The question asks for the correct formula for the transpose of the product of two matrices, \(A\) and \(B\).
Step 2: Key Formula or Approach:
This question tests a fundamental property of matrix transposes known as the "reversal rule" for products. The rule states that the transpose of a product of matrices is the product of their transposes in the reverse order.
\[ (AB)^T = B^T A^T \]
Step 3: Detailed Explanation:
Let's consider why the order must be reversed. This can be understood by looking at the element in the \(i\)-th row and \(j\)-th column of the matrices.
The element at position \((i, j)\) of the transpose of a matrix \(M\), denoted \((M^T)_{ij}\), is the same as the element at position \((j, i)\) of the original matrix \(M\), i.e., \((M^T)_{ij} = M_{ji}\).
So, the element at position \((i, j)\) of \((AB)^T\) is the element at position \((j, i)\) of \(AB\).
The \((j, i)\)-th element of \(AB\) is the dot product of the \(j\)-th row of \(A\) and the \(i\)-th column of \(B\).
Now, consider \(B^T A^T\). The \((i, j)\)-th element of this product is the dot product of the \(i\)-th row of \(B^T\) and the \(j\)-th column of \(A^T\).
The \(i\)-th row of \(B^T\) is the \(i\)-th column of \(B\), and the \(j\)-th column of \(A^T\) is the \(j\)-th row of \(A\).
Thus, the \((i, j)\)-th element of \(B^T A^T\) is the dot product of the \(j\)-th row of \(A\) and the \(i\)-th column of \(B\), which is exactly the same as the \((i, j)\)-th element of \((AB)^T\).
Step 4: Final Answer:
The value of \((AB)^T\) in matrix algebra is \(B^T A^T\).