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Problem 11: Solve the following systems of homogeneous equations by Gaussian elimination method

 Problem 11: Solve the following systems of homogeneous equations by Gaussian elimination method:

\begin{align*} x_1+x_2+x_3=0\\ x_1+2x_2+x_3=0\\ x_1+x_2+2x_3=0 \end{align*}

Solution:

Consider the augmented matrix of this system and apply row operations. \[\left[\begin{array}{c|c} A & B \end{array} \right] = \begin{align*} \left[\begin{array}{rrr|r} 1 & 1 & 1& 0 \\ 1 & 2& 1 & 0 \\ 1 & 1 & 2 & 0 \\ \end{array}\right] \end{align*}\] \[\xrightarrow[R2\rightarrow R_2-R_1]{R3\rightarrow R_3-R_1} \left[\begin{array}{rrr|r} 1 & 1 & 1& 0 \\ 0 & 1& 1 & 0 \\ 0 & 0 & 1 & 0 \\ \end{array}\right]\]

For Gaussian elimination we write down the equations corresponding to REF. Then we get

\begin{align*} x_1+x_2+x_3=0\\ x_2+x_3=0\\ x_3=0 \end{align*}

By solving we get Solution set S is \begin{align*} x_1=x_2=x_3=0. \end{align*}

The solution is trivial solution

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