\( \left[\begin{array}{ccc}1 & 4 & 3 \\ 2 & 1 & 6 \\ -1 & 2 & 0\end{array}\right] \) compute \( A+A^{\top} \) and \( A-A^{\top} \)

To compute \( A + A^{\top} \) and \( A - A^{\top} \), we first need to find the transpose of matrix \( A \). The matrix \( A \) is given as: \[ A = \left[\begin{array}{ccc}1 & 4 & 3 \\ 2 & 1 & 6 \\ -1 & 2 & 0\end{array}\right] \] The transpose of \( A \), denoted \( A^{\top} \), is obtained by swapping rows and columns: \[ A^{\top} = \left[\begin{array}{ccc}1 & 2 & -1 \\ 4 & 1 & 2 \\ 3 & 6 & 0\end{array}\right] \] Now we can compute \( A + A^{\top} \): \[ A + A^{\top} = \left[\begin{array}{ccc}1 & 4 & 3 \\ 2 & 1 & 6 \\ -1 & 2 & 0\end{array}\right] + \left[\begin{array}{ccc}1 & 2 & -1 \\ 4 & 1 & 2 \\ 3 & 6 & 0\end{array}\right] = \left[\begin{array}{ccc}1+1 & 4+2 & 3+(-1) \\ 2+4 & 1+1 & 6+2 \\ -1+3 & 2+6 & 0+0\end{array}\right] \] \[ = \left[\begin{array}{ccc}2 & 6 & 2 \\ 6 & 2 & 8 \\ 2 & 8 & 0\end{array}\right] \] Next, we compute \( A - A^{\top} \): \[ A - A^{\top} = \left[\begin{array}{ccc}1 & 4 & 3 \\ 2 & 1 & 6 \\ -1 & 2 & 0\end{array}\right] - \left[\begin{array}{ccc}1 & 2 & -1 \\ 4 & 1 & 2 \\ 3 & 6 & 0\end{array}\right] = \left[\begin{array}{ccc}1-1 & 4-2 & 3-(-1) \\ 2-4 & 1-1 & 6-2 \\ -1-3 & 2-6 & 0-0\end{array}\right] \] \[ = \left[\begin{array}{ccc}0 & 2 & 4 \\ -2 & 0 & 4 \\ -4 & -4 & 0\end{array}\right] \] Thus, the results are: \[ A + A^{\top} = \left[\begin{array}{ccc}2 & 6 & 2 \\ 6 & 2 & 8 \\ 2 & 8 & 0\end{array}\right] \] \[ A - A^{\top} = \left[\begin{array}{ccc}0 & 2 & 4 \\ -2 & 0 & 4 \\ -4 & -4 & 0\end{array}\right] \]