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Chapter 1: Q35E (page 1)

Construct a \(3 \times 3\) nonzero matrix Asuch that the vector \(\left[ {\begin{array}{*{20}{c}}1\\1\\1\end{array}} \right]\) is a solution of \(A{\bf{x}} = 0\).

Short Answer

Expert verified

The solution set in matrix form is \(A = \left[ {\begin{array}{*{20}{c}}{ - {a_2} - {a_3}}&{{a_2}}&{{a_3}}\\{ - {b_2} - {b_3}}&{{b_2}}&{{b_3}}\\{ - {c_2} - {c_3}}&{{c_2}}&{{c_3}}\end{array}} \right]\).

Step by step solution

01

Construct a \(3 \times 3\) nonzero matrix A

Since the order of matrix A is \(3 \times 3\), so matrix A consists of 3 rows and 3 columns as shown below:

Let \(A = \left[ {\begin{array}{*{20}{c}}{{a_1}}&{{a_2}}&{{a_3}}\\{{b_1}}&{{b_2}}&{{b_3}}\\{{c_1}}&{{c_2}}&{{c_3}}\end{array}} \right]\).

02

Write the matrix equation \(A{\bf{x}} = 0\)

Consider matrix \(A = \left[ {\begin{array}{*{20}{c}}{{a_1}}&{{a_2}}&{{a_3}}\\{{b_1}}&{{b_2}}&{{b_3}}\\{{c_1}}&{{c_2}}&{{c_3}}\end{array}} \right]\), and the given vector \({\bf{x}} = \left[ {\begin{array}{*{20}{c}}1\\1\\1\end{array}} \right]\).

In the matrix equation form, it can be written as shown below:

\(\begin{array}{c}A{\bf{x}} = 0\\\left[ {\begin{array}{*{20}{c}}{{a_1}}&{{a_2}}&{{a_3}}\\{{b_1}}&{{b_2}}&{{b_3}}\\{{c_1}}&{{c_2}}&{{c_3}}\end{array}} \right]\left[ {\begin{array}{*{20}{c}}1\\1\\1\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}0\\0\\0\end{array}} \right]\end{array}\)

03

Solve the matrix equation \(A{\bf{x}} = 0\)

\(\begin{array}{c}1 \cdot \left[ {\begin{array}{*{20}{c}}{{a_1}}\\{{b_1}}\\{{c_1}}\end{array}} \right] + 1 \cdot \left[ {\begin{array}{*{20}{c}}{{a_2}}\\{{b_2}}\\{{c_2}}\end{array}} \right] + 1 \cdot \left[ {\begin{array}{*{20}{c}}{{a_3}}\\{{b_3}}\\{{c_3}}\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}0\\0\\0\end{array}} \right]\\\left[ {\begin{array}{*{20}{c}}{{a_1} + {a_2} + {a_3}}\\{{b_1} + {b_2} + {b_3}}\\{{c_1} + {c_2} + {c_3}}\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}0\\0\\0\end{array}} \right]\end{array}\)

Thus, the solution set is shown below:

\(\begin{array}{l}{a_1} + {a_2} + {a_3} = 0\\{b_1} + {b_2} + {b_3} = 0\\{c_1} + {c_2} + {c_3} = 0\end{array}\)

Or, it can also be represented as shown below:

\[\begin{array}{l}{a_1} = - {a_2} - {a_3}\\{b_1} = - {b_2} - {b_3}\\{c_1} = - {c_2} - {c_3}\end{array}\]

04

Construct the matrix in the solution set form

The matrix can be represented by using \(A = \left[ {\begin{array}{*{20}{c}}{{a_1}}&{{a_2}}&{{a_3}}\\{{b_1}}&{{b_2}}&{{b_3}}\\{{c_1}}&{{c_2}}&{{c_3}}\end{array}} \right]\) as shown below:

\(A = \left[ {\begin{array}{*{20}{c}}{ - {a_2} - {a_3}}&{{a_2}}&{{a_3}}\\{ - {b_2} - {b_3}}&{{b_2}}&{{b_3}}\\{ - {c_2} - {c_3}}&{{c_2}}&{{c_3}}\end{array}} \right]\)

Thus, the solution set in matrix form is \(A = \left[ {\begin{array}{*{20}{c}}{ - {a_2} - {a_3}}&{{a_2}}&{{a_3}}\\{ - {b_2} - {b_3}}&{{b_2}}&{{b_3}}\\{ - {c_2} - {c_3}}&{{c_2}}&{{c_3}}\end{array}} \right]\) and it satisfies \(A{\bf{x}} = 0\).

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Most popular questions from this chapter

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.

Explain why a set \(\left\{ {{{\mathop{\rm v}\nolimits} _1},{{\mathop{\rm v}\nolimits} _2},{{\mathop{\rm v}\nolimits} _3},{{\mathop{\rm v}\nolimits} _4}} \right\}\) in \({\mathbb{R}^5}\) must be linearly independent when \(\left\{ {{{\mathop{\rm v}\nolimits} _1},{{\mathop{\rm v}\nolimits} _2},{{\mathop{\rm v}\nolimits} _3}} \right\}\) is linearly independent and \({{\mathop{\rm v}\nolimits} _4}\) is not in Span \(\left\{ {{{\mathop{\rm v}\nolimits} _1},{{\mathop{\rm v}\nolimits} _2},{{\mathop{\rm v}\nolimits} _3}} \right\}\).

Let \(A\) be a \(3 \times 3\) matrix with the property that the linear transformation \({\bf{x}} \mapsto A{\bf{x}}\) maps \({\mathbb{R}^3}\) into \({\mathbb{R}^3}\). Explain why transformation must be one-to-one.

Write the reduced echelon form of a \(3 \times 3\) matrix A such that the first two columns of Aare pivot columns and

\(A = \left( {\begin{aligned}{*{20}{c}}3\\{ - 2}\\1\end{aligned}} \right) = \left( {\begin{aligned}{*{20}{c}}0\\0\\0\end{aligned}} \right)\).

Find all the polynomials of degree鈮�2[a polynomial of the formf(t)=a+bt+ct2] whose graph goes through the points (1,3)and(2,6),such that f'(1)=1[wheref'(t)denotes the derivative].
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