# Let M be a transformation matrix B &#x2192;<!-- \rightarrow --> B &#x2032; </m

Let $M$ be a transformation matrix $B\to {B}^{\prime }$
discovered that ${M}^{-1}$ is the opposite transformation.
What makes it true?
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Marisol Morton
If $X$ is a matrix of a vector $x$ in the basis $B$ then its matrix $X\prime$ in $B\prime$ is
${X}^{\prime }=MX$
but in this case we have
$X={M}^{-1}{X}^{\prime }$
hence ${M}^{-1}$ is the transformation matrix ${B}^{\prime }\to B$
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Ellen Chang
I'm not sure what "opposite transformations" are, but they sound like inverse transformations and that's the definition of ${M}^{-1}$'s relationship to $M$