Q.49E 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: Q.49E (page 40)

Determine whether the statements that follow are true or false, and justify your answer.
49.If the linear system $A \overset{鈬赌}{x} = \overset{鈫�}{b}$ has a unique solution and the linear system $A \overset{鈬赌}{x} = \overset{鈬赌}{c}$ is consistent, then the linear systemlocalid="1659962306596" $A \overset{鈬赌}{x} = \overset{鈫�}{b} + \overset{鈬赌}{c}$ must have a unique solution.

Short Answer

Expert verified

The statement, 鈥淚f the linear system $A \overset{鈬赌}{x} = \overset{鈫�}{b}$ has a unique solution and the linear system is $A \overset{鈬赌}{x} = \overset{鈬赌}{c}$ consistent, then the linear system $A \overset{鈬赌}{x} = \overset{鈫�}{b} + \overset{鈬赌}{c}$ must have a unique solution.鈥� is true.

Step by step solution

Consider the condition

A consistent system will always have a unique solution.

If $A \overset{鈬赌}{x} = \overset{鈫�}{b}$ has a unique solution, then, the matrix A must have full rank.

As, $A \overset{鈬赌}{x} = \overset{鈬赌}{c}$ is consistent, thus, it must have unique solution.

Thus, $A \overset{鈬赌}{x} = \overset{鈫�}{b} + \overset{鈬赌}{c}$ must have unique solution.

Conclusion

Hence, the linear system $A \overset{鈬赌}{x} = \overset{鈫�}{b} + \overset{鈬赌}{c}$ will have unique solution for $A \overset{鈬赌}{x} = \overset{鈫�}{b}$ having a unique solution and the linear system $A \overset{鈬赌}{x} = \overset{鈬赌}{c}$ being consistent.

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Short Answer

Step by step solution

Consider the condition

Conclusion

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