Math

QuestionCheck if the system has a nontrivial solution using minimal row operations:
8x14x2+20x3=08x14x214x3=016x1+8x2+28x3=0 \begin{array}{r} 8 x_{1}-4 x_{2}+20 x_{3}=0 \\ -8 x_{1}-4 x_{2}-14 x_{3}=0 \\ 16 x_{1}+8 x_{2}+28 x_{3}=0 \end{array}
Options: A. Impossible to determine. B. Nontrivial solution. C. Only trivial solution.

Studdy Solution

STEP 1

Assumptions1. We have a system of three linear equations with three variables x1x1, xx, and x3x3. . A nontrivial solution means at least one of the variables x1x1, xx, or x3x3 is not equal to zero.
3. A trivial solution means all variables x1x1, xx, and x3x3 are equal to zero.

STEP 2

We will use the method of Gaussian elimination to find the solution of the system. First, let's write down the augmented matrix of the system.
[8420084140168280]\begin{bmatrix} 8 & -4 &20 &0\\-8 & -4 & -14 &0\\16 &8 &28 &0\end{bmatrix}

STEP 3

We can simplify the system by dividing the first row by.
[21508140168280]\begin{bmatrix} 2 & -1 &5 &0\\-8 & - & -14 &0\\16 &8 &28 &0\end{bmatrix}

STEP 4

Next, we can add4 times the first row to the second row.
[2100860168280]\begin{bmatrix} 2 & -1 & &0\\0 & -8 &6 &0\\16 &8 &28 &0\end{bmatrix}

STEP 5

We can simplify the system by dividing the second row by -8.
[2150010.750168280]\begin{bmatrix} 2 & -1 &5 &0\\0 &1 & -0.75 &0\\16 &8 &28 &0\end{bmatrix}

STEP 6

Next, we can subtract8 times the first row from the third row.
[2150010.750016120]\begin{bmatrix} 2 & -1 &5 &0\\0 &1 & -0.75 &0\\0 &16 & -12 &0\end{bmatrix}

STEP 7

Finally, we can subtract16 times the second row from the third row.
[2150010.7500000]\begin{bmatrix} 2 & -1 &5 &0\\0 &1 & -0.75 &0\\0 &0 &0 &0\end{bmatrix}

STEP 8

From the final matrix, we can see that the third equation is 0=00=0, which is always true. This means that the system is dependent and has an infinite number of solutions, including the nontrivial ones.
So, the system has a nontrivial solution.
The correct answer is B. The system has a nontrivial solution.

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