What is matrix anyway?
Well, I don't know the answer anyway, but I will come up a simple[non-official] answer for you.
Matrix is like a box and there are bunch of stuff inside the box
$\text{Identity}$ $\begin{bmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{bmatrix} \left[ \begin{array}{c} a \\ b \\ c \end{array} \right] = \left[ \begin{array}{c} a \\ b \\ c \end{array} \right]$ $\text{Scalar}$ $\begin{bmatrix} x & 0 & 0\\ 0 & y & 0\\ 0 & 0 & z \end{bmatrix} \left[ \begin{array}{c} a \\ b \\ c \end{array} \right] = \left[ \begin{array}{c} ax \\ by \\ cz \end{array} \right]$ $\text{Translation}$ $\begin{bmatrix} 1 & 0 & 0 & x\\ 0 & 1 & 0 & y\\ 0 & 0 & 1 & z\\ 0 & 0 & 0 & 1 \end{bmatrix} \left[ \begin{array}{c} a \\ b \\ c \\ d \end{array} \right] = \left[ \begin{array}{c} a + x \\ b + y \\ c + z \\ d \end{array} \right]$ $\text{Rotation}$ $M_{z}(\theta)=\begin{bmatrix} cos(\theta) & -sin(\theta) & 0\\ sin(\theta) & cos(\theta) & 0\\ 0 & 0 & 1 \end{bmatrix} \text{ rotate around the z-axis in x-y plane}$ $M_{y}(\theta)=\begin{bmatrix} cos(\theta) & sin(\theta) & 0\\ 0 & 1 & 0 \\ -sin(\theta) & cos(\theta) & 0 \end{bmatrix} \text{ rotate around the y-axis in x-z plane}$ $M_{x}(\theta)=\begin{bmatrix} 1 & 0 & 0 \\ 0 & cos(\theta) & -sin(\theta)\\ 0 & sin(\theta)& cos(\theta) \end{bmatrix} \text{ rotate around the x-axis in y-z plane}$
$\textbf{Transform normal vector using transpose inverse matrix}$ $\mbox{Let } \mathbf{n} \,, \mathbf{u} \in \mathbb{R}^n \,, \mathbf{M} \mbox{ is affine transformnation that transforms } \mathbf{u}$ $\mathbf{A} \mbox{ is affine transformation that transforms the normal vector } \mathbf{n}$
$\mathbf{n} \mbox{ is normal vector to } \mathbf{u}$
\begin{aligned} & \Rightarrow \mathbf{n}^{T} \mathbf{u} = \mathbf{0} \\ & \Rightarrow (\mathbf{A}\mathbf{n})^{T} \mathbf{M}\mathbf{u} = \mathbf{0} \\ & \Rightarrow (\mathbf{A}\mathbf{n})^{T} \mathbf{M} = \mathbf{n}^{T}\\ & \Rightarrow (\mathbf{A}\mathbf{n})^{T} = \mathbf{n}^{T}\mathbf{M}^{-1} \\ & \Rightarrow (\mathbf{A}\mathbf{n}) = \mathbf{M}^{-T} \mathbf{n}\\ & \Rightarrow \mathbf{A} = \mathbf{M}^{-T} \nonumber \\ \end{aligned}