Question

Determine whether the statements that follow are true or false, and justify your answer.

49.If the linear system$\mathit{A}\stackrel{\mathbf{\rightharpoonup }}{\mathbf{x}}\mathbf{=}\overrightarrow{\mathbf{b}}$ has a unique solution and the linear system$\mathit{A}\stackrel{\mathbf{\rightharpoonup }}{\mathbf{x}}\mathbf{=}\stackrel{\mathbf{\rightharpoonup }}{\mathbf{c}}$ is consistent, then the linear systemlocalid="1659962306596" $\mathit{A}\stackrel{\mathbf{\rightharpoonup }}{\mathbf{x}}\mathbf{=}\overrightarrow{\mathbf{b}}\mathbf{+}\stackrel{\mathbf{\rightharpoonup }}{\mathbf{c}}$ must have a unique solution.

Short answer

The statement, “If the linear system $A\stackrel{\rightharpoonup }{x}=\overrightarrow{b}$ has a unique solution and the linear system is$A\stackrel{\rightharpoonup }{x}=\stackrel{\rightharpoonup }{c}$consistent, then the linear system$A\stackrel{\rightharpoonup }{x}=\overrightarrow{b}+\stackrel{\rightharpoonup }{c}$ must have a unique solution.” is true.

Step-by-step solution

Consider the condition

A consistent system will always have a unique solution.

If$A\stackrel{\rightharpoonup }{x}=\overrightarrow{b}$has a unique solution, then, the matrix A must have full rank.

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

Thus,$A\stackrel{\rightharpoonup }{x}=\overrightarrow{b}+\stackrel{\rightharpoonup }{c}$must have unique solution.

Conclusion 

Hence, the linear system $A\stackrel{\rightharpoonup }{x}=\overrightarrow{b}+\stackrel{\rightharpoonup }{c}$ will have unique solution for$A\stackrel{\rightharpoonup }{x}=\overrightarrow{b}$ having a unique solution and the linear system$A\stackrel{\rightharpoonup }{x}=\stackrel{\rightharpoonup }{c}$ being consistent.