About a proof of Cayley-Hamilton theorem in "Linear Algebra" by Ichiro Satake.

Solution 1:

To summarize: yes, one can get away with proving that $A$ and $B_k$ commute to the price of explaining how to understand a polynomial with matrix coefficients, e.g. all powers of $x$ to the left, prove that, in this case, for the first line in (7) to hold it is enough that $x$ and $A$ commute and, at the end, remark on that $p(A)$ could be understood as an ordinary polynomial (i.e. no need to collect all powers of $A$ to the left any more) as $A$ and $c_kI$ also commute. It is quite close to what the first (direct algebraic) proof in Wiki does implicitly.

It feels that to prove commutation of $A$ and $B_k$ (simply by comparing the coefficients in the first and the second lines in (7)) and forget the trouble is somewhat easier.