Q41E Determine whether the statements... [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 1: Q41E (page 40)

Determine whether the statements that follow are true or false, and justify your answer.
41.The linear system $A \overset{鈫�}{x} = \overset{鈫�}{b}$ is consistent if (and only if) $r a n k (A) = r a n k [A | \overset{鈫�}{b}]$ .

Short Answer

Expert verified

The statement, 鈥淭he linear system $A \overset{鈫�}{x} = \overset{鈫�}{b}$ is consistent if (and only if) $r a n k (A) = r a n k [A | \overset{鈬赌}{b}]$ .鈥� is true.

Step by step solution

Consider the condition.

Consider a different condition.

For $r a n k (A) < r a n k [A | \overset{鈬赌}{b}]$ , the system will become inconsistent.

As the system has to be consistent, so, the condition $r a n k (A) < r a n k [A | \overset{鈬赌}{b}]$ does not work.

Reduced row echelon form

For a random system, the reduced row echelon form of a system is usually $[\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}]$ , which will have a unique solution.

The system will be consistent, if and only if the number of equations equals the number of unknowns.

As the condition $r a n k (A) = r a n k [A | \overset{鈫�}{b}]$ satisfies the system $A \overset{鈫�}{x} = \overset{鈫�}{b}$ to be consistent, thus, the statement is correct.

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91影视

Determine whether the statements that follow are true or false, and justify your answer.
41.The linear system $A \overset{鈫�}{x} = \overset{鈫�}{b}$ is consistent if (and only if) $r a n k (A) = r a n k [A | \overset{鈫�}{b}]$ .

Short Answer

Step by step solution

Consider the condition.

Reduced row echelon form

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91影视

Determine whether the statements that follow are true or false, and justify your answer.41.The linear systemAx鈫�=b鈫� is consistent if (and only if) rank(A)=rank[A|b鈫�].

Short Answer

Step by step solution

Consider the condition.

Reduced row echelon form

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Calculus

Geometry

Probability and Statistics

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Decision Maths

Study anywhere. Anytime. Across all devices.

Determine whether the statements that follow are true or false, and justify your answer.
41.The linear system $A \overset{鈫�}{x} = \overset{鈫�}{b}$ is consistent if (and only if) $r a n k (A) = r a n k [A | \overset{鈫�}{b}]$ .