= Solution
The <adjugate matrix> is the transpose of the cofactor matrix, and it satisfies
$$
M\operatorname{adj}(M)=\operatorname{adj}(M)M=(\det M)I.
$$
Put $M=tI-A$ and substitute the two given expansions into this identity:
$$
(tI-A)\sum_{i=0}^{n-1}B_it^{n-1-i}
=\sum_{j=0}^nc_jt^{n-j}I.
$$
Comparing coefficients of powers of $t$ gives
$$
\boxed{B_0=c_0I=I,
\qquad B_i=AB_{i-1}+c_iI\quad(1\leq i\leq n-1),
\qquad -AB_{n-1}=c_nI}.
$$
Here $c_0=1$ because the <characteristic polynomial> is monic.
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