Q52E For A=[012024036148], find a vec... [FREE SOLUTION]

91影视

Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 2: Q52E (page 100)

For $A = [\begin{matrix} 0 & 1 & 2 \\ 0 & 2 & 4 \\ 0 & 3 & 6 \\ 1 & 4 & 8 \end{matrix}]$ , find a vector $\overset{鈫�}{b} in {鈩�}^{4}$ such that the system is $A \overset{鈫�}{x} = \overset{鈫�}{b}$ inconsistent.

Short Answer

Expert verified

For $A = [\begin{matrix} 0 & 1 & 2 \\ 0 & 2 & 4 \\ 0 & 3 & 6 \\ 1 & 4 & 8 \end{matrix}]$ , there exists a vector $\overset{鈫�}{b} = [\begin{matrix} 0 \\ 0 \\ 1 \\ 1 \end{matrix}] i n {鈩�}^{4}$ such that the system $A \overset{鈫�}{x} = \overset{鈫�}{b}$ is inconsistent.

Step by step solution

Mentioning the concept

If we have a matrix A of order n x m and an augmented matrix $[A \overset{鈫�}{b}]$ then the system of linear equations $A \overset{鈫�}{x} = \overset{鈫�}{b}$ has inconsistent solution if (and only if)

$r a n k (A) 鈮� r a n k [A \overset{鈫�}{b}]$

Taking a vector b⇀ such that the system Ax→=b→ is inconsistent

Let us take $\overset{鈬赌}{b} = [\begin{matrix} 0 \\ 0 \\ 1 \\ 1 \end{matrix}]$ .

Then the that the systemrole="math" localid="1659350117618" $A \overset{鈫�}{x} = \overset{鈫�}{b}$ is inconsistent if (and only if) $r a n k (A) 鈮� r a n k [A \overset{鈫�}{b}]$ .

Here we have $[\begin{matrix} 0 & 1 & 2 \\ 0 & 2 & 4 \\ 0 & 3 & 6 \\ 1 & 4 & 8 \end{matrix}]$ .

Finding rank (A) and rank [Ab→]

Find rank as:

$[A \overset{鈫�}{b}] = [\begin{matrix} 0 & 1 & 2 \\ 0 & 2 & 4 \\ 0 & 3 & 6 \\ 1 & 4 & 8 \end{matrix} \begin{matrix} 0 \\ 0 \\ 1 \\ 1 \end{matrix}] (R_{1} 鈫� R_{4}) 鈬� [A \overset{鈫�}{b}] = [\begin{matrix} 1 & 4 & 8 \\ 0 & 2 & 4 \\ 0 & 3 & 6 \\ 0 & 1 & 2 \end{matrix} \begin{matrix} 1 \\ 0 \\ - 3 \\ 0 \end{matrix}] (\begin{matrix} R_{3} = 3 R_{2} - 2 R_{3} \\ R_{4} = R_{2} - 2 R_{4} \end{matrix})$

Since $r a n k (A) = 2 a n d r a n k (A b) = 3, 鈬� r a n k (A) 鈮� r a n k (A \overset{鈫�}{b})$

Thus,the given system $A \overset{鈫�}{x} = \overset{鈫�}{b}$ is inconsistent.

Final Answer

For $A = [\begin{matrix} 0 & 1 & 2 \\ 0 & 2 & 4 \\ 0 & 3 & 6 \\ 1 & 4 & 8 \end{matrix}]$ , there exists a vector $\overset{鈫�}{b} = [\begin{matrix} 0 \\ 0 \\ 1 \\ 1 \end{matrix}]$ in ${鈩�}^{4}$ such that the system $A \overset{鈫�}{x} = \overset{鈫�}{b}$ is inconsistent.

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91影视

For $A = [\begin{matrix} 0 & 1 & 2 \\ 0 & 2 & 4 \\ 0 & 3 & 6 \\ 1 & 4 & 8 \end{matrix}]$ , find a vector $\overset{鈫�}{b} in {鈩�}^{4}$ such that the system is $A \overset{鈫�}{x} = \overset{鈫�}{b}$ inconsistent.

Short Answer

Step by step solution

Mentioning the concept

Taking a vector b⇀ such that the system Ax→=b→ is inconsistent

Finding rank (A) and rank [Ab→]

Final Answer

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91影视

For A=[012024036148], find a vector b鈫�in鈩�4such that the system isAx鈫�=b鈫� inconsistent.

Short Answer

Step by step solution

Mentioning the concept

Taking a vector b⇀ such that the system Ax→=b→ is inconsistent

Finding rank (A) and rank [Ab→]

Final Answer

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Probability and Statistics

Geometry

Logic and Functions

Theoretical and Mathematical Physics

Mechanics Maths

Study anywhere. Anytime. Across all devices.

For $A = [\begin{matrix} 0 & 1 & 2 \\ 0 & 2 & 4 \\ 0 & 3 & 6 \\ 1 & 4 & 8 \end{matrix}]$ , find a vector $\overset{鈫�}{b} in {鈩�}^{4}$ such that the system is $A \overset{鈫�}{x} = \overset{鈫�}{b}$ inconsistent.