1. **State the problem:** We are given matrix $$A=\begin{bmatrix} 1 & 2 & 3 \\ -4 & -4 & -4 \\ 5 & 6 & -7 \end{bmatrix}$$ and we need to find its transpose, denoted as $$A^T$$. 2. **Recall the definition:** The transpose of a matrix $$A$$ is obtained by swapping rows and columns. That means the first row of $$A$$ becomes the first column of $$A^T$$, the second row becomes the second column, etc. 3. **Write down rows of $$A$$:** - Row 1: $$[1, 2, 3]$$ - Row 2: $$[-4, -4, -4]$$ - Row 3: $$[5, 6, -7]$$ 4. **Form columns of $$A^T$$ from rows of $$A$$:** - Column 1 of $$A^T$$: $$[1, -4, 5]^T$$ - Column 2 of $$A^T$$: $$[2, -4, 6]^T$$ - Column 3 of $$A^T$$: $$[3, -4, -7]^T$$ 5. **Write the transpose matrix:** $$A^T=\begin{bmatrix} 1 & -4 & 5 \\ 2 & -4 & 6 \\ 3 & -4 & -7 \end{bmatrix}$$ **Final answer:** $$A^T=\begin{bmatrix} 1 & -4 & 5 \\ 2 & -4 & 6 \\ 3 & -4 & -7 \end{bmatrix}$$