2. Calcule o valor de S: \[ S=\left|\begin{array}{ccc}2 & -1 & 1 \\ 1 & 1 & 1 \\ -2 & 1 & 1\end{array}\right|+\left|\begin{array}{lll}0 & 1 & 4 \\ 0 & 2 & 5 \\ 0 & 3 & 6\end{array}\right|+\left|\begin{array}{ccc}-1 & 2 & 3 \\ 5 & 0 & -2 \\ 1 & 6 & 4\end{array}\right| \]

Para calcular o valor de S, precisamos calcular os determinantes das três matrizes dadas e somá-los. Vamos calcular os determinantes das matrizes: 1. Para a primeira matriz: \[ S_1 = \left|\begin{array}{ccc}2 & -1 & 1 \\ 1 & 1 & 1 \\ -2 & 1 & 1\end{array}\right| \] 2. Para a segunda matriz: \[ S_2 = \left|\begin{array}{lll}0 & 1 & 4 \\ 0 & 2 & 5 \\ 0 & 3 & 6\end{array}\right| \] 3. Para a terceira matriz: \[ S_3 = \left|\begin{array}{ccc}-1 & 2 & 3 \\ 5 & 0 & -2 \\ 1 & 6 & 4\end{array}\right| \] Vamos calcular os determinantes das matrizes e somá-los para obter o valor de S. Matrices by following steps: - step0: Evaluate the determinant: \(\left[\begin{array}{rrr}{2}&{-1}&{1}\\{1}&{1}&{1}\\{-2}&{1}&{1}\end{array}\right]\) - step1: Use the appropriate notation: \(\left|\begin{array}{cc}{2}&{-1}&{1}\\{1}&{1}&{1}\\{-2}&{1}&{1}\end{array}\right|\) - step2: Use cofactor expansion along a row: \(2\times \left|\begin{array}{cc}{1}&{1}\\{1}&{1}\end{array}\right|-\left(-\left|\begin{array}{cc}{1}&{1}\\{-2}&{1}\end{array}\right|\right)+1\times \left|\begin{array}{cc}{1}&{1}\\{-2}&{1}\end{array}\right|\) - step3: Evaluate the determinant: \(2\times 0-\left(-3\right)+1\times 3\) - step4: Multiply: \(0-\left(-3\right)+3\) - step5: Rewrite the expression: \(0+3+3\) - step6: Remove 0: \(3+3\) - step7: Add the numbers: \(6\) Calculate the determinant of the matrix \( \left[\begin{array}{lll}0 & 1 & 4 \\ 0 & 2 & 5 \\ 0 & 3 & 6\end{array}\right] \). Matrices by following steps: - step0: Evaluate the determinant: \(\left[\begin{array}{rrr}{0}&{1}&{4}\\{0}&{2}&{5}\\{0}&{3}&{6}\end{array}\right]\) - step1: Use the appropriate notation: \(\left|\begin{array}{cc}{0}&{1}&{4}\\{0}&{2}&{5}\\{0}&{3}&{6}\end{array}\right|\) - step2: Use cofactor expansion along a column: \(0\times \left|\begin{array}{cc}{2}&{5}\\{3}&{6}\end{array}\right|-0\times \left|\begin{array}{cc}{1}&{4}\\{3}&{6}\end{array}\right|+0\times \left|\begin{array}{cc}{1}&{4}\\{2}&{5}\end{array}\right|\) - step3: Multiply: \(0-0+0\) - step4: Rewrite the expression: \(0+0+0\) - step5: Add the numbers: \(0\) Calculate the determinant of the matrix \( \left[\begin{array}{ccc}-1 & 2 & 3 \\ 5 & 0 & -2 \\ 1 & 6 & 4\end{array}\right] \). Matrices by following steps: - step0: Evaluate the determinant: \(\left[\begin{array}{rrr}{-1}&{2}&{3}\\{5}&{0}&{-2}\\{1}&{6}&{4}\end{array}\right]\) - step1: Use the appropriate notation: \(\left|\begin{array}{cc}{-1}&{2}&{3}\\{5}&{0}&{-2}\\{1}&{6}&{4}\end{array}\right|\) - step2: Use cofactor expansion along a row: \(-5\times \left|\begin{array}{cc}{2}&{3}\\{6}&{4}\end{array}\right|+0\times \left|\begin{array}{cc}{-1}&{3}\\{1}&{4}\end{array}\right|-\left(-2\times \left|\begin{array}{cc}{-1}&{2}\\{1}&{6}\end{array}\right|\right)\) - step3: Evaluate the determinant: \(-5\left(-10\right)+0-\left(-2\left(-8\right)\right)\) - step4: Evaluate: \(-\left(-50\right)+0-16\) - step5: Rewrite the expression: \(50+0-16\) - step6: Remove 0: \(50-16\) - step7: Subtract the numbers: \(34\) O valor de S é a soma dos determinantes das três matrizes: \[ S = S_1 + S_2 + S_3 = 6 + 0 + 34 = 40 \] Portanto, o valor de S é 40.