I am solving a system of linear equations $Ax= y$ in $n$ dimensions and I have found that in my case a matrix $A$ is upper diagonal with zeros on the main diagonal, while the next diagonal has non-zero values. So, iirc from my LA courses, it means that $\mathrm{rank}(A) = n-1$ and if I provide a proper extra condition, it will make this system determined.
In my case $x$ is a vector of values a certain function takes at predefined points and $A$ corresponds to the numerical integration scheme, more precisely a trapezoid rule. I hence thought that I may have a freedom in assigning boundary values to $x$, say making $x_n = 0$. However in such case it would constitute to replacing the lower row of the matrix form of $A$ with the $e_n$ basis vector, i.e. $a_{nj} = 0$ for $j < n$ as before but now $a_{nn} = 1$. However, since $a_{00}$ is still $0$, the rank of $A$ is still $n-1$. What am I missing here and how should I properly reguarize $A$?