The normal distribution of random variable $x$ is $$p(x)=Norm_x[Ay+b,\Sigma]$$, the mean $\mu=Ay+b$ is a function of another variable $y$.
My problem is how to derive the normal distribution of $y$, which means take $x$ as parameter.
Here is what I tried, \begin{align*}p(x)&=\frac{1}{\sqrt{2\pi}|\Sigma|^{1/2}}\exp\left[-(x-Ay-b)^T\Sigma^{-1}(x-Ay-b)\right]\\ &=\frac{1}{\sqrt{2\pi}|\Sigma|^{1/2}}\exp\left[-(A^{-1}x-y-A^{-1}b)^TA^T\Sigma^{-1}A(A^{-1}x-y-A^{-1}b)\right]\\ &=\frac{1}{\sqrt{2\pi}|\Sigma|^{1/2}}\exp\left[-(y-A^{-1}(x-b))^TA^T\Sigma^{-1}A(y-A^{-1}(x-b))\right]\\ &=\frac{1}{|A|^2}\frac{|A|^2}{\sqrt(2\pi)|\Sigma|^{1/2}}\exp\left[-(y-A^{-1}(x-b))^TA^T\Sigma^{-1}A(y-A^{-1}(x-b))\right]\end{align*}, alright, now I think for variable $y$, the mean and covariance are $$\mu'=A^{-1}(x-b)$$$$\Sigma'=(A^T\Sigma^{-1}A)^{-1}$$, if so I have $$Norm_x[Ay+b,\Sigma]=\kappa\cdot Norm_y[\mu',\Sigma']$$, where $\kappa=\frac{1}{|A|^2}$.
BUT, the result in the book I'm reading is different, which is
Is there anything wrong with my derivation?
Observe that in your book $$A'=\Sigma'A^T\Sigma^{-1}=(A^T\Sigma^{-1}A)^{-1}A^T\Sigma^{-1}=A^{-1}\Sigma\underbrace{(A^T)^{-1}A^T}_{I}\Sigma^{-1}=A^{-1}\underbrace{\Sigma\Sigma^{-1}}_{I}=A^{-1}$$ and $$b'=-(A^T\Sigma^{-1}A)^{-1}A^T\Sigma^{-1}=-A'b=-A^{-1}b$$ so that $$ \mu'=A'x+b'=A^{-1}x-A^{-1}b=A^{-1}(x-b) $$ wich is your result.