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Chapter 1: Q5E (page 1)

In Exercises 5 and 6, follow the method of Examples 1 and 2 to write the solution set of the given homogeneous system in parametric vector form.

5. \(\begin{aligned}{c}{x_1} + 3{x_2} + {x_3} = 0\\ - 4{x_1} - 9{x_2} + 2{x_3} = 0\\ - 3{x_2} - 6{x_3} = 0\end{aligned}\)

Short Answer

Expert verified

The general solution in the parametric vector form is represented as \(x = {x_3}\left( {\begin{array}{*{20}{c}}5\\{ - 2}\\1\end{array}} \right)\).

Step by step solution

01

Convert the given system of equations into an augmented matrix

Anaugmented matrix for a system of equations is a matrix of numbers in which eachrowrepresents the constants from one equation, and eachcolumn 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 \({x_1} + 3{x_2} + {x_3} = 0, - 4{x_1} - 9{x_2} + 2{x_3} = 0\) and \( - 3{x_2} - 6{x_3} = 0\) is represented as:

\(\left[ {\begin{array}{*{20}{c}}1&3&1&0\\{ - 4}&{ - 9}&2&0\\0&{ - 3}&{ - 6}&0\end{array}} \right]\)

02

Apply row operation

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

Perform the sum of \(4\) times row 1 and row 2 at row 2.

\(\left[ {\begin{array}{*{20}{c}}1&3&1&0\\0&3&6&0\\0&{ - 3}&{ - 6}&0\end{array}} \right]\)

03

Apply row operation

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

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

\(\left[ {\begin{array}{*{20}{c}}1&3&1&0\\0&3&6&0\\0&0&0&0\end{array}} \right]\)

04

Apply row operation

Perform an elementary row operationto produce the third augmented matrix.

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

\(\left[ {\begin{array}{*{20}{c}}1&0&{ - 5}&0\\0&3&6&0\\0&0&0&0\end{array}} \right]\)

05

Apply row operation

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

Multiply row 2 by \(\frac{1}{3}\).

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

06

Convert the matrix into the equation

To obtain the solution of the system of equations, you have to convert the augmented matrix into the system of equations again.

Write the obtained matrix \(\left[ {\begin{array}{*{20}{c}}1&0&{ - 5}&0\\0&1&2&0\\0&0&0&0\end{array}} \right]\) into the equation notation.

\(\begin{array}{c}{x_1} - 5{x_3} = 0\\{x_2} + 2{x_3} = 0\end{array}\)

Thus, \({x_1} = 5{x_3},{x_2} = - 2{x_3}\), and \({x_3}\) is a free variable.

07

Determine the general solution in the parametric vector form

The general solution of \(Ax = 0\) in the parametric vector form can be represented as:

\(\begin{array}{c}x = \left[ {\begin{array}{*{20}{c}}{{x_1}}\\{{x_2}}\\{{x_3}}\end{array}} \right]\\ = \left[ {\begin{array}{*{20}{c}}{5{x_3}}\\{ - 2{x_3}}\\{{x_3}}\end{array}} \right]\\ = {x_3}\left[ {\begin{array}{*{20}{c}}5\\{ - 2}\\1\end{array}} \right]\end{array}\)

Thus, the general solution in the parametric vector form is \(x = {x_3}\left[ {\begin{array}{*{20}{c}}5\\{ - 2}\\1\end{array}} \right].\)

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Most popular questions from this chapter

Find the elementary row operation that transforms the first matrix into the second, and then find the reverse row operation that transforms the second matrix into the first.

30.\(\left[ {\begin{array}{*{20}{c}}1&3&{ - 4}\\0&{ - 2}&6\\0&{ - 5}&9\end{array}} \right]\), \(\left[ {\begin{array}{*{20}{c}}1&3&{ - 4}\\0&1&{ - 3}\\0&{ - 5}&9\end{array}} \right]\)

In Exercises 32, find the elementary row operation that transforms the first matrix into the second, and then find the reverse row operation that transforms the second matrix into the first.

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

Question: Determine whether the statements that follow are true or false, and justify your answer.

14: rank.|111123136|=3

Consider a dynamical system x鈫�(t+1)=Ax鈫�(t) with two components. The accompanying sketch shows the initial state vector x鈫�0and two eigen vectors 蠀1鈫�鈥�鈥�and鈥�鈥�蠀2鈫� of A (with eigen values 位1鈫�and位2鈫� respectively). For the given values of 位1鈫�and位2鈫�, draw a rough trajectory. Consider the future and the past of the system.

位1鈫�=1,位2鈫�=0.9

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.

24.

a. Elementary row operations on an augmented matrix never change the solution set of the associated linear system.

b. Two matrices are row equivalent if they have the same number of rows.

c. An inconsistent system has more than one solution.

d. Two linear systems are equivalent if they have the same solution set.
See all solutions

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