1. **State the problem:** Find the transpose of the matrix $$A = \begin{bmatrix} 1 & 2 & 1 \\ -3 & -2 & 9 \\ -5 & 7 & -3 \end{bmatrix}$$ 2. **Recall the definition:** The transpose of a matrix $A$, denoted $A^T$, is obtained by swapping its rows and columns. That means the element at position $(i,j)$ in $A$ becomes the element at position $(j,i)$ in $A^T$. 3. **Apply the rule:** - The first row of $A$ becomes the first column of $A^T$: $[1, 2, 1] \to \begin{bmatrix}1 \\ 2 \\ 1\end{bmatrix}$ - The second row of $A$ becomes the second column of $A^T$: $[-3, -2, 9] \to \begin{bmatrix}-3 \\ -2 \\ 9\end{bmatrix}$ - The third row of $A$ becomes the third column of $A^T$: $[-5, 7, -3] \to \begin{bmatrix}-5 \\ 7 \\ -3\end{bmatrix}$ 4. **Write the transpose matrix:** $$A^T = \begin{bmatrix} 1 & -3 & -5 \\ 2 & -2 & 7 \\ 1 & 9 & -3 \end{bmatrix}$$ 5. **Final answer:** The transpose of matrix $A$ is $$\boxed{\begin{bmatrix} 1 & -3 & -5 \\ 2 & -2 & 7 \\ 1 & 9 & -3 \end{bmatrix}}$$