Can one evaluate this expression with only two matrix multiplications?
Matrices $A$ and $B$ are of the same square size, say $4\times 4$. We want to evaluate:
$A^2+A*B*A$
I would write $A^2$ as $A*A$, but then $A*B*A$ is not equivalent to $A*A*B$, so I would need three multiplications, $A*A$ , $A*B$ and the product of $A*B$ multiplied with $A$.
What am I missing?
To summarize the discussion in the comments:
We can write $$A+ABA=A(\mathbb 1 + B)A$$ where $\mathbb 1$ denotes the relevant identity matrix.
After the trivial matrix addition, we now only require two multiplies, namely $A(\mathbb 1+B)$ and then $(A(\mathbb 1+B))A$.