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Chapter 1: Q11E (page 1)

Given \(A\) and \(b\) in Exercises 11 and 12, write the augmented matrix for the linear system that corresponds to the matrix equation \(Ax = b\). Then solve the system and write the solution as a vector.

11. \({\mathop{\rm A}\nolimits} = \left( {\begin{array}{*{20}{c}}1&2&4\\0&1&5\\{ - 2}&{ - 4}&{ - 3}\end{array}} \right),b = \left( {\begin{array}{*{20}{c}}{ - 2}\\2\\9\end{array}} \right)\)

Short Answer

Expert verified

The solution in the vector form is \(x = \left( {\begin{array}{*{20}{c}}0\\{ - 3}\\1\end{array}} \right)\).

Step by step solution

01

Writing the matrix in the augmented form

The augmented form of the matrix is

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

02

Applyingthe row operation

Perform an elementary row operationto produce the first augmented matrix.

The sum of 2 times row one and row threeis written in row three.

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

03

Applying the row operation

Perform an elementary row operationto produce the second augmented matrix.

Multiply row three by \(\frac{1}{5}\).

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

04

Applying the row operation

Perform an elementary row operationto produce the fourth augmented matrix.

The sum of \( - 4\) times row three and row oneis written in row one, and the sum of \( - 5\) times rowsthree and row two is written in row two.

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

05

Applying the row operation

Perform an elementary row operationto produce the fifth augmented matrix.

The sum of \( - 2\) times row two and row one is written in row one.

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

06

Converting the matrix into the equation form

To obtain the solution of the vector equation, convert the augmented matrix into vector equations.

The obtained matrix,\(\left( {\begin{array}{*{20}{c}}1&0&0&0\\0&1&0&{ - 3}\\0&0&1&1\end{array}} \right)\),in the equation notation is shown below.

\(\begin{array}{l}{x_1} = 0\\{x_2} = - 3\\{x_3} = 1\end{array}\)

07

Writing the solution of the system as a vector

The solution of the system as a vector is shown below.

\(\begin{array}{c}x = \left( {\begin{array}{*{20}{c}}{{x_1}}\\{{x_2}}\\{{x_3}}\end{array}} \right)\\ = \left( {\begin{array}{*{20}{c}}0\\{ - 3}\\1\end{array}} \right)\end{array}\)

Thus, the solution in the vector form is \(x = \left( {\begin{array}{*{20}{c}}0\\{ - 3}\\1\end{array}} \right)\).

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Most popular questions from this chapter

In Exercises 7-12, describe all solutions of \(Ax = 0\) in parametric vector form, where \(A\) is row equivalent to the given matrix.

12. \(\left( {\begin{array}{*{20}{c}}1&5&2&{ - 6}&9&0\\0&0&1&{ - 7}&4&{ - 8}\\0&0&0&0&0&1\\0&0&0&0&0&0\end{array}} \right)\)

Consider an economy with three sectors, Chemicals & Metals, Fuels & Power and Machinery. Chemicals sell \(30\% \) of its output to fuels and \(50\% \) to Machinery and retains the rest. Fuels sells \(80\% \) of its output to chemicals and \(10\% \) to Machinery and retains the rest. Machinery sells \(40\% \) to chemicals and \(40\% \) to Fuels and retains the rest.

a. Construct the exchange table for the economy

b. Develop a system of equations that leads to prices at which each sector鈥檚 income matches its expanses. Then write the augmented matrix that can be row reduced to find these prices.

c. \(\left[ M \right]\)Find a set of equilibrium prices when the price for the machinery output is 100 units.

Give a geometric description of span \(\left\{ {{v_1},{v_2}} \right\}\) for the vectors \({{\mathop{\rm v}\nolimits} _1} = \left[ {\begin{array}{*{20}{c}}8\\2\\{ - 6}\end{array}} \right]\) and \({{\mathop{\rm v}\nolimits} _2} = \left[ {\begin{array}{*{20}{c}}{12}\\3\\{ - 9}\end{array}} \right]\).

Construct a \(3 \times 3\) matrix, not in echelon form, whose columns span \({\mathbb{R}^3}\). Show that the matrix you construct has the desired property.

Suppose \(a,b,c,\) and \(d\) are constants such that \(a\) is not zero and the system below is consistent for all possible values of \(f\) and \(g\). What can you say about the numbers \(a,b,c,\) and \(d\)? Justify your answer.

28. \(\begin{array}{l}a{x_1} + b{x_2} = f\\c{x_1} + d{x_2} = g\end{array}\)
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