If we let $x∼N_3(\mu, \Sigma)$ with $\mu^T=(\mu_1,\mu_2,\mu_3)$ and $\Sigma=\begin{pmatrix}\sigma ^2&\sigma ^2\rho &\sigma \:^2\rho \:\\ \sigma \:^2\rho \:&\sigma \:^2&\sigma \:^2\rho \:\\ \sigma \:^2\rho \:&\sigma \:^2\rho \:&\sigma \:^2\end{pmatrix}$.
How do I find the conditional distribution of $(x_2, x_3),$ given $x_1$? (i.e. $(x_2,x_3|x_1)$)
I tried to use something like $$\left(\begin{matrix} x_2+x_3 \\ x_1 \end{matrix}\right)=\begin{pmatrix}0&1&1\\ 1&0&0\end{pmatrix}\begin{pmatrix}x_1\\ x_2\\ x_3\end{pmatrix}=AX$$ Which would then give $$A\mu=\begin{pmatrix}0&1&1\\ 1&0&0\end{pmatrix}\begin{pmatrix}\mu _1\\ \mu _2\\ \mu _3\end{pmatrix}=\left(\begin{matrix} \mu_2+\mu_3 \\ \mu_1 \end{matrix}\right)$$
And I would also need a covariance matrix in the form of $$A\Sigma A'=\begin{pmatrix}0&1&1\\ 1&0&0\end{pmatrix}\begin{pmatrix}\sigma ^2&\sigma ^2\rho &\sigma \:^2\rho \:\\ \sigma \:^2\rho \:&\sigma \:^2&\sigma \:^2\rho \:\\ \sigma \:^2\rho \:&\sigma \:^2\rho \:&\sigma \:^2\end{pmatrix}\begin{pmatrix}0&1\\ 1&0\\ 1&0\end{pmatrix}=something$$
However, after this I am stuck and I am not sure if what I have done is correctly or what I should do next.
