I have the following system of equations \begin{align} \begin{bmatrix} {p_x}\\{p_y}\\{p_z} \end{bmatrix} = \begin{bmatrix} m_{11} & m_{12} & m_{13}\\ m_{21} & m_{22} & m_{23}\\ m_{31} & m_{32} & m_{33} \end{bmatrix}\begin{bmatrix} a_x\\a_y\\a_z \end{bmatrix} \end{align} which in Einstein notation, it simplifies to \begin{align*} p_i = m_{ij}a_j\quad i,j = 1,2,3 \end{align*} It is assumed that $a_i$, which is the input vector, and $m_{ij}$, which is the transfer function, have Gaussian distributions \begin{align*} a_i \sim \mathcal{N}(\overline{a_i}, \sigma_i^2)\\ m_{ij} \sim \mathcal{N}(\overline{m_{ij}}, \sigma_{ij}^2) \end{align*}
For a transfer function $m_{ij}$, I have three independent measurements \begin{align*} \begin{bmatrix} {p_x}_1\\{p_y}_1\\{p_z}_1\\{p_x}_2\\{p_y}_2\\{p_z}_2\\{p_x}_3\\{p_y}_3\\{p_z}_3 \end{bmatrix} = \begin{bmatrix} m_{11} & m_{12} & m_{13}&0&0&0&0&0&0\\ m_{21} & m_{22} & m_{23}&0&0&0&0&0&0\\ m_{31} & m_{32} & m_{33}&0&0&0&0&0&0\\ 0&0&0&m_{11} & m_{12} & m_{13}&0&0&0\\ 0&0&0&m_{21} & m_{22} & m_{23}&0&0&0\\ 0&0&0&m_{31} & m_{32} & m_{33}&0&0&0\\ 0&0&0&0&0&0&m_{11} & m_{12} & m_{13}\\ 0&0&0&0&0&0&m_{21} & m_{22} & m_{23}\\ 0&0&0&0&0&0&m_{31} & m_{32} & m_{33}\\ \end{bmatrix}\begin{bmatrix} {a_x}_1\\{a_y}_1\\{a_z}_1\\{a_x}_2\\{a_y}_2\\{a_z}_2\\{a_x}_3\\{a_y}_3\\{a_z}_3 \end{bmatrix} \end{align*} or in matrix form \begin{align*} P = MA \end{align*}
Given $\overline{a_i}$, $\sigma_i^2$, $\overline{m_{ij}}$, and $\sigma_{ij}^2$, how can I find the vector $P$, having 9 components, with the following three constraints ($P_{12}$ and $P_{23}$ are given as well) \begin{align} \det\left(\begin{bmatrix} {p_x}_1 & {p_y}_1 & {p_z}_1\\ {p_x}_2 & {p_y}_2 & {p_z}_2\\ {p_x}_3 & {p_y}_3 & {p_z}_3 \end{bmatrix}\right) = 0 \end{align}
\begin{align} \left({p_x}_1-{p_x}_2\right)^2 + \left({p_y}_1-{p_y}_2\right)^2 + \left({p_z}_1-{p_z}_2\right)^2 = P_{12}^2\\ \left({p_x}_2-{p_x}_3\right)^2 + \left({p_y}_2-{p_y}_3\right)^2 + \left({p_z}_2-{p_z}_3\right)^2 = P_{23}^2 \end{align}
I would appreciate any hint for solving this optimization problem. I hope my problem explanation makes sense and I look forward to any suggestions.