If $A$ is a non-singular $n\times n$ matrix, $B$ is an $n\times p$ matrix, and $C$ is a $p\times n$ matrix (where $1\le p \ll n$), how does one prove that the complexity of $$D=A^{-1}(BC)$$ is $\frac{8}{3}N^3+2pn^2+O(n)$ flops?I have no idea where to start. It is assumed that we do not compute $A^{-1}$ directly, but, instead, we compute the $LU$-factorization of $A$, and use $L$ and $U$ to compute the product $A^{-1}(BC)$ using the method of solving of multiple right-hand sides (whatever this means).I would appreciate some help with this.
Proving complexity of computing product of matrices
complexity theory;time complexity;linear algebra