 Home

Codex Wiki

List of universities

University of Cambridge

Course of the University of Cambridge

Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2019

Paper 1

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.

Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / ib / Paper 1 / 9F / a / Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / ib / Paper 1 / 9F / a / Solution

 Ancestors (11)

 Discussion (0)

 Articles by others on the same topic (0)

Past exam of the mathematics course of the University of Cambridge / 2019 / ib / Paper 1 / 9F / a / Solution

 Ancestors (11)

 Discussion (0)  Subscribe (1)

 Articles by others on the same topic (0)

 Discussion (0)