TheoremVault | Commuting Matrices Product

Be $A,B$ and $C$ three matrices of order n. Proof that if $A$ commutes with $B$ and $A$ commutes with $C$ , then $A$ commutes with the $BC$ .

Show Solution

As we know $AB = BA$ so we can multiply each side by C on the right:

$ABC = BAC$

And as we know $AC = CA$ so we can substitute in the previous equation:

$ABC = BCA$

Therefore, A commutes with the product of B and C.