1. **State the problem:** We are given the matrix $$A = \begin{bmatrix} 2 & 3 \\ 1 & 4 \end{bmatrix}$$ and need to find its transpose. 2. **Formula for transpose:** The transpose of a matrix $$A$$, denoted $$A^T$$, is obtained by swapping its rows and columns. Formally, if $$A = [a_{ij}]$$, then $$A^T = [a_{ji}]$$. 3. **Apply the formula:** For matrix $$A$$, $$A = \begin{bmatrix} 2 & 3 \\ 1 & 4 \end{bmatrix}$$ - The first row of $$A$$ is $$[2, 3]$$ which becomes the first column of $$A^T$$. - The second row of $$A$$ is $$[1, 4]$$ which becomes the second column of $$A^T$$. 4. **Write the transpose:** $$A^T = \begin{bmatrix} 2 & 1 \\ 3 & 4 \end{bmatrix}$$ 5. **Explanation:** Transposing flips the matrix over its diagonal, turning rows into columns and vice versa. **Final answer:** $$A^T = \begin{bmatrix} 2 & 1 \\ 3 & 4 \end{bmatrix}$$