$\mathbf{X}\in N(\mathbf{0},\mathbf{\Lambda})$ Find the conditional distribution $P(X_{1},X_{2}|X_{3}=x)$

Question

Let $\mathbf{X}\in N(\mathbf{0},\mathbf{\Lambda})$ Find the conditional distribution $P(X_{1},X_{2}|X_{3}=x)$ \begin{equation*} \begin{split} \mathbf{\Lambda}=\left(\begin{array}{ccc} v_{11} & v_{12} &v_{13} \\ v_{21} & v_{22} &v_{23}\\ v_{31} & v_{32} & v_{33} \end{array}\right) \end{split} \end{equation*} From Wikipedia, https://en.wikipedia.org/wiki/Multivariate_normal_distribution#Conditional_distributions \begin{equation*} \begin{split} \mathbf{x_{1}}, \mathbf{x_{2}}\\ \mathbf{x}=\left[\begin{array}{c} \mathbf{x_{1}} \\ \mathbf{x_{2}} \end{array}\right]\quad \text{with sizes}\quad \left[\begin{array}{c} q\times 1 \\ (N-q)\times 1 \end{array}\right]\\ \mathbf{\mu}=\left[\begin{array}{c} \mathbf{\mu_{1}} \\ \mathbf{\mu_{2}} \end{array}\right]\quad \text{with sizes} \quad \left[\begin{array}{c} q\times 1 \\ (N-q)\times 1 \end{array}\right]\\ \mathbf{\Sigma}=\left[\begin{array}{cc} \mathbf{\Sigma_{11}} &\mathbf{\Sigma_{12}} \\ \mathbf{\Sigma_{21}} & \mathbf{\Sigma_{22}} \end{array}\right]\quad \text{with sizes}\quad \left[\begin{array}{cc} q\times q & q\times (N-q)\\ (N-q)\times q & (N-q)\times(N-q) \end{array}\right]\\ \end{split} \end{equation*} \begin{equation*} \begin{split} \mathbf{\mu}=\mu_{1}+\Sigma_{12}\Sigma_{22}^{-1}\left(a-\mu_{2}\right)\\ \mathbf{\Sigma}=\Sigma_{11}-\Sigma_{12}\Sigma_{22}^{-1}\Sigma_{21} \end{split} \end{equation*} In this case $N=3$ and $q=2$. I partition $\Lambda$. \begin{equation*} \begin{split} \mathbf{\Lambda}=\left(\begin{array}{cc|c} v_{11} & v_{12} &v_{13} \\ v_{21} & v_{22} &v_{23}\\ \hline v_{31} & v_{32} & v_{33} \end{array}\right) \end{split} \end{equation*} \begin{equation*} \begin{split} \Sigma_{11}=\left[\begin{array}{cc} v_{11} & v_{12} \\ v_{21} & v_{22} \end{array}\right]\\ \Sigma_{12}=\left[\begin{array}{c} v_{13} \\ v_{23} \end{array}\right]\\ \Sigma_{21}=\left[\begin{array}{cc} v_{31} & v_{32} \end{array}\right]\\ \Sigma_{22}=\left[\begin{array}{c} v_{33} \end{array}\right]\\ \Sigma_{22}^{-1}=\frac{1}{v_{33}}\left[\begin{array}{c} v_{33} \end{array}\right] \end{split} \end{equation*} \begin{equation*} \begin{split} \mathbf{\mu}=\mu_{1}+\frac{1}{v_{33}}\left[\begin{array}{c} v_{13} \\ v_{23} \end{array}\right]\left[\begin{array}{c} v_{33} \end{array}\right]\left[\begin{array}{c} x-\mu_{3} \end{array}\right]\\ \mathbf{\Sigma}=\left[\begin{array}{cc} v_{11} & v_{12} \\ v_{21} & v_{22} \end{array}\right]-\frac{1}{v_{33}}\left[\begin{array}{c} v_{13} \\ v_{23} \end{array}\right]\left[\begin{array}{c} v_{33} \end{array}\right]\left[\begin{array}{cc} v_{31} & v_{32} \end{array}\right] \end{split} \end{equation*} The matrix multiplication in the last two expressions are $2\times 1\cdot 1\times 1$ and I cannot evaluate this.Where did I go wrong in trying to find the distribution $P(X_{1},X_{2}|X_{3}=x)$?