This is a very basic question, but unfortunately I cannot find an answer to it. Let $A$ and $B$ be square invertible $n \times n$ matrices. Let $\vec c$ be an $n \times 1$ vector.
If we have
$ A\vec c = B\vec c$
does $A = B$ ? Does $\vec c$ cancel in a sense?
If it depends on some conditions, under what general conditions is it true?
Thank you in advance.
$\endgroup$ 71 Answer
$\begingroup$$\begin{bmatrix}1&0\\0&1\end{bmatrix}\begin{bmatrix}1\\0\end{bmatrix} = \begin{bmatrix}1&0\\0&2\end{bmatrix}\begin{bmatrix}1\\0\end{bmatrix}=\begin{bmatrix}1\\0\end{bmatrix}$
Both $\begin{bmatrix}1&0\\0&1\end{bmatrix}$ and $\begin{bmatrix}1&0\\0&2\end{bmatrix}$ are invertible, however they are nonequal.
This acts as a counterexample to the claim above. $Ac=Bc$ with $A$ and $B$ invertible does not imply that $A=B$
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