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We know that if we want to reflect any point over an origin, i.e. $ O\left(0, 0\right) $, we can use matrix transformation like this $$ \left(\begin{matrix}x' \\ y'\end{matrix}\right) = \left(\begin{matrix}-1 & 0 \\ 0 & -1\end{matrix}\right)\left(\begin{matrix}x \\ y\end{matrix}\right) = \left(\begin{matrix}-x \\ -y\end{matrix}\right). $$ But, what if we reflect any point over another point $ M\left(a, b\right) $ with $ a, b \ne 0 $?

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This would not generally be a linear transformation since $(0,0)$ would not map to itself via a reflection over a non-origin point. So you will not be able to do this with a single matrix multiplication.

I would suggest translating the coordinate system so that the reflection point is at the new origin; then reflect; and then translate back.

For example, to translate $(x,y)$ over $(5,7)$, I would do

$\left(\begin{array}{c} x' \\ y' \end{array}\right)=\left(\begin{array}{c} x-5 \\ y-7 \end{array}\right)\left(\begin{array}{cc} -1&0 \\0&-1 \end{array}\right)+ \left(\begin{array}{c} 5 \\ 7 \end{array}\right)$

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