Vaughan Fuentes
09/19/2024 · Senior High School
2. Calcule o valor de S : \[ S=\left|\begin{array}{ccc}2 & -1 & 1 \\ 1 & 1 & 1 \\ -2 & 1 & 1\end{array}\right|+\left|\begin{array}{lll}0 & 1 & 4 \\ 0 & 2 & 5 \\ 0 & 3 & 6\end{array}\right|+\left|\begin{array}{ccc}-1 & 2 & 3 \\ 5 & 0 & -2 \\ 1 & 6 & 4\end{array}\right| \]
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Step-by-step Solution
Para calcular o valor de \( S \), precisamos calcular os determinantes das três matrizes dadas e somá-los.
Vamos calcular cada determinante separadamente:
1. Para a primeira matriz:
\[ \left|\begin{array}{ccc}2 & -1 & 1 \\ 1 & 1 & 1 \\ -2 & 1 & 1\end{array}\right| \]
2. Para a segunda matriz:
\[ \left|\begin{array}{lll}0 & 1 & 4 \\ 0 & 2 & 5 \\ 0 & 3 & 6\end{array}\right| \]
3. Para a terceira matriz:
\[ \left|\begin{array}{ccc}-1 & 2 & 3 \\ 5 & 0 & -2 \\ 1 & 6 & 4\end{array}\right| \]
Vamos calcular cada determinante e somá-los para encontrar 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:
\(6\)
Calculou o determinante da primeira matriz e obteve o valor de 6.
Agora, vamos calcular o determinante da segunda matriz:
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:
\(0\)
Calculou o determinante da segunda matriz e obteve o valor de 0.
Agora, vamos calcular o determinante da terceira matriz:
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:
\(34\)
Calculou o determinante da terceira matriz e obteve o valor de 34.
Agora, vamos somar os determinantes para encontrar o valor de \( S \):
\[ S = 6 + 0 + 34 = 40 \]
Portanto, o valor de \( S \) é 40.
Quick Answer
Para encontrar o valor de \( S \), calcule os determinantes das três matrizes e some-os. O valor de \( S \) é 40.
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