To solve the problem and compute \( B^T A^T \), we begin by finding the transpose of matrices \( A \) and \( B \). Given:
| \( A = \begin{bmatrix} 2 & -1 \\ -7 & 4 \end{bmatrix} \) | \( B = \begin{bmatrix} 4 & 1 \\ 7 & 2 \end{bmatrix} \) |
Step 1: Compute the transpose of matrix \( A \):
\( A^T = \begin{bmatrix} 2 & -7 \\ -1 & 4 \end{bmatrix} \)
Step 2: Compute the transpose of matrix \( B \):
\( B^T = \begin{bmatrix} 4 & 7 \\ 1 & 2 \end{bmatrix} \)
Step 3: Multiply \( B^T \) and \( A^T \) to find \( B^T A^T \):
The product \( B^T A^T \) is calculated as:
\( B^T A^T = \begin{bmatrix} 4 & 7 \\ 1 & 2 \end{bmatrix} \times \begin{bmatrix} 2 & -7 \\ -1 & 4 \end{bmatrix} \)
The element in the first row and first column of the result is:
\((4 \times 2) + (7 \times -1) = 8 - 7 = 1\)
The element in the first row and second column of the result is:
\((4 \times -7) + (7 \times 4) = -28 + 28 = 0\)
The element in the second row and first column of the result is:
\((1 \times 2) + (2 \times -1) = 2 - 2 = 0\)
The element in the second row and second column of the result is:
\((1 \times -7) + (2 \times 4) = -7 + 8 = 1\)
Thus, \( B^T A^T = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \), which is an identity matrix.
Conclusion: The matrix \( B^T A^T \) is an identity matrix, confirming the correct answer is "an identity matrix".