Given the matrices \( A \) and \( B \) shown below, find \( -2 A+6 B \). \[ A=\left[\begin{array}{ccc}-1 & 0 & 5 \\ -1 & -1 & -6 \\ -5 & -6 & 0\end{array}\right] \quad B=\left[\begin{array}{ccc}1 & 2 & -4 \\ 4 & 1 & 3 \\ 2 & 1 & -3\end{array}\right] \]

To find \( -2 A + 6 B \), we need to first calculate \( -2 A \) and \( 6 B \) separately, and then add them together. Starting with matrix \( A \): \[ A = \begin{bmatrix} -1 & 0 & 5 \\ -1 & -1 & -6 \\ -5 & -6 & 0 \end{bmatrix} \] Calculating \( -2 A \): \[ -2 A = -2 \begin{bmatrix} -1 & 0 & 5 \\ -1 & -1 & -6 \\ -5 & -6 & 0 \end{bmatrix} = \begin{bmatrix} 2 & 0 & -10 \\ 2 & 2 & 12 \\ 10 & 12 & 0 \end{bmatrix} \] Next, we calculate matrix \( B \): \[ B = \begin{bmatrix} 1 & 2 & -4 \\ 4 & 1 & 3 \\ 2 & 1 & -3 \end{bmatrix} \] Calculating \( 6 B \): \[ 6 B = 6 \begin{bmatrix} 1 & 2 & -4 \\ 4 & 1 & 3 \\ 2 & 1 & -3 \end{bmatrix} = \begin{bmatrix} 6 & 12 & -24 \\ 24 & 6 & 18 \\ 12 & 6 & -18 \end{bmatrix} \] Now we add \( -2 A \) and \( 6 B \): \[ -2 A + 6 B = \begin{bmatrix} 2 & 0 & -10 \\ 2 & 2 & 12 \\ 10 & 12 & 0 \end{bmatrix} + \begin{bmatrix} 6 & 12 & -24 \\ 24 & 6 & 18 \\ 12 & 6 & -18 \end{bmatrix} \] Calculating the sum element-wise: \[ \begin{bmatrix} 2 + 6 & 0 + 12 & -10 - 24 \\ 2 + 24 & 2 + 6 & 12 + 18 \\ 10 + 12 & 12 + 6 & 0 - 18 \end{bmatrix} \] This results in: \[ \begin{bmatrix} 8 & 12 & -34 \\ 26 & 8 & 30 \\ 22 & 18 & -18 \end{bmatrix} \] Thus, the final result of \( -2 A + 6 B \) is: \[ \boxed{\begin{bmatrix} 8 & 12 & -34 \\ 26 & 8 & 30 \\ 22 & 18 & -18 \end{bmatrix}} \]