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Chapter 1: Q1E (page 1)

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

Expert verified

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

Step by step solution

01

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:

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

02

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.

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

03

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.

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

04

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.

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

05

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.

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Most popular questions from this chapter

Suppose the system below is consistent for all possible values of \(f\) and \(g\). What can you say about the coefficients \(c\) and \(d\)? Justify your answer.

27. \(\begin{array}{l}{x_1} + 3{x_2} = f\\c{x_1} + d{x_2} = g\end{array}\)

Let \(A = \left[ {\begin{array}{*{20}{c}}1&0&{ - 4}\\0&3&{ - 2}\\{ - 2}&6&3\end{array}} \right]\) and \(b = \left[ {\begin{array}{*{20}{c}}4\\1\\{ - 4}\end{array}} \right]\). Denote the columns of \(A\) by \({{\mathop{\rm a}\nolimits} _1},{a_2},{a_3}\) and let \(W = {\mathop{\rm Span}\nolimits} \left\{ {{a_1},{a_2},{a_3}} \right\}\).

  1. Is \(b\) in \(\left\{ {{a_1},{a_2},{a_3}} \right\}\)? How many vectors are in \(\left\{ {{a_1},{a_2},{a_3}} \right\}\)?
  2. Is \(b\) in \(W\)? How many vectors are in W.
  3. Show that \({a_1}\) is in W.[Hint: Row operations are unnecessary.]

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In Exercises 23 and 24, key statements from this section are either quoted directly, restated slightly (but still true), or altered in some way that makes them false in some cases. Mark each statement True or False, and justify your answer. (If true, give the approximate location where a similar statement appears, or refer to a de铿乶ition or theorem. If false, give the location of a statement that has been quoted or used incorrectly, or cite an example that shows the statement is not true in all cases.) Similar true/false questions will appear in many sections of the text.

23.

a. Every elementary row operation is reversible.

b. A \(5 \times 6\)matrix has six rows.

c. The solution set of a linear system involving variables \({x_1},\,{x_2},\,{x_3},........,{x_n}\)is a list of numbers \(\left( {{s_1},\, {s_2},\,{s_3},........,{s_n}} \right)\) that makes each equation in the system a true statement when the values \ ({s_1},\, {s_2},\, {s_3},........,{s_n}\) are substituted for \({x_1},\,{x_2},\,{x_3},........,{x_n}\), respectively.

d. Two fundamental questions about a linear system involve existence and uniqueness.

In Exercise 2, compute \(u + v\) and \(u - 2v\).

2. \(u = \left[ {\begin{array}{*{20}{c}}3\\2\end{array}} \right]\), \(v = \left[ {\begin{array}{*{20}{c}}2\\{ - 1}\end{array}} \right]\).
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