Interesting matrices

Using the given definitions for the matrix $A$ , we can evaluate it as follows:

$A=\begin{pmatrix}1&1&1\\ -a_{12}&2&2\\ -a_{13}&-a_{23}&3\end{pmatrix}=\begin{pmatrix}1&1&1\\ -1&2&2\\ -1&-2&3\end{pmatrix}\ldots (i)$

But from the problem, we have that,

$A=\begin{pmatrix}y-x&1&x-z\\ -1&y-z&2\\ x-y&-2&x+z\end{pmatrix}\ldots (ii)$

From $(i)$ and $(ii)$ , we can see that both the matrices are equal and as such their entries must be equal accordingly. We obtain the following system of equations by comparing both sides of the equation that can be obtained from $(i)$ and $(ii)$ :

$y-x=1~,~x-z=1~,~x+z=3~,~y-z=2$

Solving this system of equations yields $x=2,y=3,z=1$ and hence the required sum is $x+y+5=2+3+1=\boxed{6}$