Commutative Property Matrices that Differ by a Constant

Question

Let $A$ be a matrix, $P$ a Polynomial in its linear factors $$P(x):=\prod_{i=1}(\lambda_i-x)$$ How does one know that the multiplication of Matrices when we calculate $P(A)$ is commutative, as the matrices to be multiplied vary ($\lambda_i$ varies)? It must be commutative, or else the order of the linear factors would matter and the same polynomial could deliver different results on the same input.

Answer 1

$A$ commutes with itself and the identity matrix (hidden behind those $\lambda_i$s) commutes with all the other matrices so it is all good really.