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Question states:

True or False?

If the linear system $A^2x =b$ is consistent, then the system $Ax=b$ is consistent as well.

I haven't found any kind of counterexamples. It seems like that the rank of $A$ is equal to the rank of $A^2$, although I wasn't able to verify this. Also, even if I could verify this, I don't get how this would imply that the answer to my question is yes.

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    As for your remark about the ranks of $A$ and $A^2$: take $A = \begin{pmatrix} 0 & 1\\ 0 & 0 \end{pmatrix}$. Then $A$ has rank $1$ but $A^2$ has rank $0$. – 2017-01-27

1 Answers 1

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If $x=x_0$ solves $A^2x=b$, then $x=x_1=Ax_0$ solves $Ax=b$.