Question

If \( A = \begin{bmatrix} 2 & -1 \\ -7 & 4 \end{bmatrix} \) and \( B = \begin{bmatrix} 4 & 1 \\ 7 & 2 \end{bmatrix} \), then \( B^T A^T \) is:

• A. null matrix
• B. an identity matrix
• C. scalar but not an identity matrix
• D. such that \( \text{Tr}(B^T A^T) = 4 \)

Hint:

\((AB)^T = B^T A^T\) — always use this property to simplify.

Answer 1

The Correct Option is B

Concept: \[ (B^T A^T) = (AB)^T \]

Step 1: Compute \(AB\) \[ A = \begin{bmatrix} 2 & -1 \\ -7 & 4 \end{bmatrix}, \quad B = \begin{bmatrix} 4 & 1 \\ 7 & 2 \end{bmatrix} \] \[ AB = \begin{bmatrix} 2\cdot4 + (-1)\cdot7 & 2\cdot1 + (-1)\cdot2 \\ -7\cdot4 + 4\cdot7 & -7\cdot1 + 4\cdot2 \end{bmatrix} \] \[ = \begin{bmatrix} 8 - 7 & 2 - 2 \\ -28 + 28 & -7 + 8 \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \]

Step 2: \[ AB = I \Rightarrow (AB)^T = I \]