How do I find specific transfer functions from a plant of a MIMO system?
For any linear time-invariant system the transfer function will be

$$ G(s) = YU^{-1} = C (sI-A)^{-1}B + D,$$

and the blank matrix in you question will be \$G(s) \in \mathcal{R}^{4 \times 6} \$, with the top element of the output matrix \$ z\$

$$z_{1} = ([C_{1i}] (sI-A)^{-1}B + D)U,$$

where \$ [C_{1i}] \$ is the row vector with the elements in the first row of \$C\$. Do notice that \$z_1(u_1,u_2,\dots,n_2) \$.

If you permute the inputs in \$U\$ it is the same as applying an invertible matrix \$ P\$ to have a \$ \hat{U} = PU\$. So, as long as \$ P\$ is just a permutation (a single 1 per line and no columns has more than a single 1), if \$G(1,1)\$ gives you a TF you will be able to find that same TF somewhere in \$ \hat{G} \$.

Moreover, if you have

$$ \begin{bmatrix}
d_1\\ 
d_2\\ 
u_1\\ 
u_2\\ 
n_1\\ 
n_2
\end{bmatrix} \rightarrow G,$$

$$ \begin{bmatrix}
u_1\\ 
u_2\\ 
d_1\\ 
d_2\\ 
n_1\\ 
n_2
\end{bmatrix} \rightarrow \hat{G},$$

The permutation is

$$ P = \begin{bmatrix}
   0  &amp; 0  &amp; 1  &amp; 0  &amp; 0  &amp; 0 \\
   0  &amp; 0  &amp; 0  &amp; 1  &amp; 0  &amp; 0 \\
   1  &amp; 0  &amp; 0  &amp; 0  &amp; 0  &amp; 0 \\
   0  &amp; 1  &amp; 0  &amp; 0  &amp; 0  &amp; 0 \\
   0  &amp; 0  &amp; 0  &amp; 0  &amp; 1  &amp; 0 \\
   0  &amp; 0  &amp; 0  &amp; 0  &amp; 0  &amp; 1
\end{bmatrix}$$

and leads to 

$$  G(1,1) = \hat{G}(1,3). $$