In Exercises 1-4, determine if the system has a nontrivial solution. Try to use a few row operations as possible.

2. \(\begin{aligned}{c}{x_1} - 3{x_2} + 7{x_3} = 0\\ - 2{x_1} + {x_2} - 4{x_3} = 0\\{x_1} + 2{x_2} + 9{x_3} = 0\end{aligned}\)

Anaugmented matrix for a system of equations is a matrix of numbers in which eachrowrepresents the constantsfrom one equation, and each column represents all thecoefficients for a single variable.

The augmented matrix \(\left( {\begin{array}{*{20}{c}}A&0\end{array}} \right)\) for the given system of equations \({x_1} - 3{x_2} + 7{x_3} = 0, - 2{x_1} + {x_2} - 4{x_3} = 0\),and \({x_1} + 2{x_2} + 9{x_3} = 0\) is represented as:

\(\left[ {\begin{array}{*{20}{c}}1&{ - 3}&7&0\\{ - 2}&1&{ - 4}&0\\1&2&9&0\end{array}} \right]\)

Perform an elementary row operation to produce the first augmented matrix.

Perform the sum of \(2\) times row 1 and row 2 at row 2.

\(\left[ {\begin{array}{*{20}{c}}1&{ - 3}&7&0\\0&{ - 5}&{10}&0\\1&2&9&0\end{array}} \right]\)

Perform an elementary row operation to produce the second augmented matrix.

Perform the sum of \( - 1\) times row 1 and row 3 at row 3.

\(\left[ {\begin{array}{*{20}{c}}1&{ - 3}&7&0\\0&{ - 5}&{10}&0\\0&5&2&0\end{array}} \right]\)

Perform an elementary row operation to produce the third augmented matrix.

Perform the sum of \(1\) times row 2 and row 3 at row 3.

\(\left[ {\begin{array}{*{20}{c}}1&{ - 3}&7&0\\0&{ - 5}&{10}&0\\0&0&{12}&0\end{array}} \right]\)

It is known that the homogeneous equation \(Ax = 0\) has a nontrivial solutionif and only if the equation has at least one free variable.

The system has anontrivial solution if a column in the coefficient matrix does not construct a pivot column and the corresponding variable is free.

\(\left[ {\begin{array}{*{20}{c}}1&{ - 3}&7&0\\0&{ - 5}&{10}&0\\0&0&{12}&0\end{array}} \right]\)

The system has only a trivial solution since it has no free variable.

Thus, the system of linear equations only has a trivial solution.