I was looking at an answer about matrixes and I found this statement:
$$ \pi_A(A)=\prod_{k=1}^n(A-\lambda_kI)=\prod_{k=1}^nS(D-\lambda_kI)S^{-1}=S\prod_{k=1}^n(D-\lambda_kI)S^{-1}=S\cdot 0\cdot S^{-1}=0 $$
The equations' context was within matrixes and eigenvectors. But, besides the question context, I don't understand how he was able to go from step 2 to step 3. Meaning, why is this true:
$\prod_{k=1}^nS(D-\lambda_kI)S^{-1}=S\prod_{k=1}^n(D-\lambda_kI)S^{-1}$
Doesn't this break a rule? And if not, which rule supports this operation?
Let's make the substitution $B_k = D-\lambda_kI$.
Then, $$ \prod_{k=1}^n (S B_k S^{-1}) = (S B_1 S^{-1})(S B_2 S^{-1})\cdots (S B_n S^{-1}) $$ $$= S B_1 (S^{-1} S) B_2 (S^{-1} S) B_3 \cdots (S^{-1} S) B_n S^{-1} $$ $$= S B_1 B_2 \cdots B_n S^{-1} $$ $$= S \left(\prod_{k=1}^n B_k\right) S^{-1}$$ $$= S \left(\prod_{k=1}^n \left(D-\lambda_k I \right)\right) S^{-1}.$$