Interesting matrices

Algebra Level 3

If \(A=(a_{ij})\) is a \(3\times 3\) matrix such that \(a_{ij}=i\) for \(i\leq j\) and \(a_{ij}=-a_{ji}\) for \(i>j,\) and \[A=\left(\begin{array}{ccc}y-x&1& x-z\\-1& y-z&2\\x-y&-2& x+z \end{array}\right),\] what is the value of \(x+y+z?\)

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