I have a matrix with some elements known and some unknown. I am using the notation $A(X)$ where $X$ are the unknown elements (not sure if relevant but I will be solving for the unknown part $X$ later). For example, $$ A(x_{11},x_{22})=\left(\begin{array}{cc} x_{11} & 1 \\ 0 & x_{22} \end{array}\right) $$ then $x_{11}$ and $x_{22}$ are unknown.
Is this good notation? Should I be adding something to the notation to make it clearer? Is there better or standard notation for such matrices?
This is fine, but a widely followed convention is to denote matrices by uppercase letters and their entries using the corresponding lowercase letters, so a slightly more conventional choice of notation would be
$$ A(a_{11},a_{22})=\left(\begin{array}{cc} a_{11} & 1 \\ 0 & a_{22} \end{array}\right)\;, $$
and you don't really need to include $a_{11},a_{22}$ as arguments unless you really want to emphasize the dependence, so
$$ A=\left(\begin{array}{cc} a_{11} & 1 \\ 0 & a_{22} \end{array}\right) $$
would be clear enough. But everyone will understand it the way you wrote it, too.