Question

What is the value of \((AB)^T\) in matrix algebra?

• A. \(A^T B^T\)
• B. \(B^T A^T\)
• C. \(AB\)
• D. \(A^T B\)

Hint:

When taking the transpose of a matrix product, {reverse the order of multiplication}: \[ (AB)^T = B^T A^T \]

Answer 1

The Correct Option is B

Concept: In matrix algebra, the transpose of a matrix is obtained by interchanging its rows and columns. If \(A\) is an \(m \times n\) matrix, then its transpose \(A^T\) is an \(n \times m\) matrix. One important property of matrix transpose is related to the product of two matrices.
Step 1: Recall the transpose property of matrix multiplication.
For any two matrices \(A\) and \(B\) whose product \(AB\) is defined, the transpose of their product satisfies: \[ (AB)^T = B^T A^T \] This means the order of multiplication reverses when taking the transpose.
Step 2: Understand the reversal of order.
Matrix multiplication is not commutative, meaning: \[ AB \neq BA \] Therefore, when taking the transpose of a product, the matrices must appear in reverse order.
Step 3: Conclusion.
Using the transpose rule for matrix multiplication: \[ (AB)^T = B^T A^T \] Hence, the correct answer is: \[ B^T A^T \]