I'm struggling to solve the below statement. I'm familiar with proving it with matrix multiplication (ex, showing $Y = X_1 - \Sigma_{12}\Sigma_{22}^{-1}X_2$ and $X_2$ are independent), but this requires different way.
Suppose $X \sim N_n(\mu, \Sigma)$ with nonsingular $\Sigma$, and partition the matrices $X,\mu, \Sigma$ as follows: $$X = \begin{bmatrix} X_1 \\ X_2 \end{bmatrix}, \mu = \begin{bmatrix} \mu_1 \\ \mu_2 \end{bmatrix}, \Sigma = \begin{bmatrix} \Sigma_{11} & \Sigma_{12} \\ \Sigma_{21} & \Sigma_{22} \end{bmatrix},$$ where $X_1 \in \mathbb R^r$ and $X_2 \in \mathbb R^{n-r}$.
Show the following statement using the pdf $f_{1|2}$ only; $$X_1|X_2 \sim N_r\left(\mu_1 + \Sigma_{12}\Sigma_{22}^{-1}(X_2 - \mu_2), \Sigma_{11} - \Sigma_{12}\Sigma_{22}^{-1}\Sigma_{21} \right)$$
That is, I have to prove the statement using $f_X(x) = (2\pi)^{-n/2}|\Sigma|^{-1/2}\text{exp}\{-\frac{1}{2}(x-\mu)^T\Sigma^{-1}(x-\mu)\}$
We have $f_{X_2}(x) := f_2(x) = (2\pi)^{-(n-r)/2}|\Sigma_{22}|^{-1/2}\text{exp}\{-\frac{1}{2}(x-\mu_2)^T\Sigma_{22}^{-1}(x-\mu_2)\}$
Also, $|\Sigma| = |\Sigma_{22}||\Sigma_{11} - \Sigma_{12}\Sigma_{22}^{-1}\Sigma_{21}|$.
Then, $$\begin{aligned} f_{1|2}(x_1|x_2) & = f_X(x_1,x_2)/f_2(x_2) \\ & = (2\pi)^{r/2}|\Sigma_{11} - \Sigma_{12}\Sigma_{22}^{-1}\Sigma_{21}|^{-1/2}\text{exp}\{-\frac{1}{2}(x-\mu)^T\Sigma^{-1}(x-\mu)\}/\text{exp}\{-\frac{1}{2}(x_2-\mu_2)^T\Sigma_{22}^{-1}(x_2-\mu_2)\} \end{aligned} $$
This is what I have done, and I need some help about how to proceed to obtain the conditional pdf of $X_1|X_2$
Any help regarding this question would be grateful. Thank you.