Question

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

41.The linear system$\mathit{A}\overrightarrow{\mathbf{x}}\mathbf{=}\overrightarrow{\mathbf{b}}$ is consistent if (and only if) $\mathit{r}\mathit{a}\mathit{n}\mathit{k}\mathbf{\left(}\mathit{A}\mathbf{\right)}\mathbf{=}\mathit{r}\mathit{a}\mathit{n}\mathit{k}\left[A|\overrightarrow{b}\right]$.

Short answer

The statement, “The linear system $A\overrightarrow{x}=\overrightarrow{b}$is consistent if (and only if) $rank\left(A\right)=rank\left[A|\stackrel{\rightharpoonup }{b}\right]$.” is true.

Step-by-step solution

Consider the condition.

Consider a different condition.

For$rank\left(A\right)<rank\left[A|\stackrel{\rightharpoonup }{b}\right]$, the system will become inconsistent.

As the system has to be consistent, so, the condition $rank\left(A\right)<rank\left[A|\stackrel{\rightharpoonup }{b}\right]$does not work.

Reduced row echelon form 

For a random system, the reduced row echelon form of a system is usually $\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$, 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$rank\left(A\right)=rank\left[A|\overrightarrow{b}\right]$ satisfies the system$A\overrightarrow{x}=\overrightarrow{b}$ to be consistent, thus, the statement is correct.