Question

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

1. \(\begin{aligned}{l}2{x_1} - 5{x_2} + 8{x_3} = 0\\ - 2{x_1} - 7{x_2} + {x_3} = 0\\4{x_1} + 2{x_2} + 7{x_3} = 0\end{aligned}\)

Short answer

The system has a nontrivial solution since \({x_3}\) is a free variable.

Step-by-step solution

Convert the given system of equations into an augmented matrix

The augmented matrix \(\left( {\begin{array}{*{20}{c}}A&0\end{array}} \right)\) for the given system of equations \(2{x_1} - 5{x_2} + 8{x_3} = 0, - 2{x_1} - 7{x_2} + {x_3} = 0\), and \(4{x_1} + 2{x_2} + 7{x_3} = 0\) is represented as:

\(\left[ {\begin{array}{*{20}{c}}2&{ - 5}&8&0\\{ - 2}&{ - 7}&1&0\\4&2&7&0\end{array}} \right]\)

Apply row operation

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

Perform the sum of 1 times row 1 and row 2 at row 2.

\(\left[ {\begin{array}{*{20}{c}}2&{ - 5}&8&0\\0&{ - 12}&9&0\\4&2&7&0\end{array}} \right]\)

Apply row operation

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

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

\(\left[ {\begin{array}{*{20}{c}}2&{ - 5}&8&0\\0&{ - 12}&9&0\\0&{12}&{ - 9}&0\end{array}} \right]\)

Apply row operation

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.

\(\left[ {\begin{array}{*{20}{c}}2&{ - 5}&8&0\\0&{ - 12}&9&0\\0&0&0&0\end{array}} \right]\)

Determine whether the given system has a nontrivial solution

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 a nontrivial solution if a column in the coefficient matrix does not construct a pivot column.

\(\left[ {\begin{array}{*{20}{c}}2&{ - 5}&8&0\\0&{ - 12}&9&0\\0&0&0&0\end{array}} \right]\)

Since \({x_3}\) is a free variable, the system has a nontrivial solution.

Thus, the system of linear equations has a nontrivial solution.