Matrix Transpose Determination
To find \( A^T \), which is the transpose of the given matrix \( A \), you need to swap the rows of \( A \) with its columns. The matrix \( A \) is:
\[ A = \begin{bmatrix} 5 & 1 \\ 2 & 3 \\ -1 & 4 \end{bmatrix} \]
So the transpose of this matrix, denoted \( A^T \), where each row of \( A \) becomes a column, will be:
\[ A^T = \begin{bmatrix} 5 & 2 & -1 \\ 1 & 3 & 4 \end{bmatrix} \]
Matching this with the options provided:
A.
\[ \begin{bmatrix} 5 & 1 \\ 2 & 3 \\ 1 & 4 \end{bmatrix} \] (incorrect)
B.
\[ \begin{bmatrix} 5 & 2 & -1 \\ 1 & 3 & 4 \end{bmatrix} \] (correct)
C.
\[ \begin{bmatrix} -5 & -2 & -3 \\ 1 & -4 \end{bmatrix} \] (incorrect)
D. Nothing above (incorrect)
The correct answer is option B.
