#adjoint-method

$A(p)$ is an invertible $n \times n$ matrix built from $n - 1$ parameters $p$, and $f(p) = \left(c^\top A(p)^{-1} b\right)^2$ for fixed vectors $b, c$. Using the adjoint (reverse-mode) method, how many linear solves with $A$ (or $A^\top$) are needed to get $f$ and the whole gradient $\nabla_p f$, whatever $n$ is?