Paper 1, Section II, A

(a) For an $n \times n$ matrix $A$ define the characteristic polynomial $\chi_{A}$ and the characteristic equation.

The Cayley-Hamilton theorem states that every $n \times n$ matrix satisfies its own characteristic equation. Verify this in the case $n=2$.

(b) Define the adjugate matrix $\operatorname{adj}(A)$ of an $n \times n$ matrix $A$ in terms of the minors of $A$. You may assume that

$A \operatorname{adj}(A)=\operatorname{adj}(A) A=\operatorname{det}(A) I$

where $I$ is the $n \times n$ identity matrix. Show that if $A$ and $B$ are non-singular $n \times n$ matrices then

$\operatorname{adj}(A B)=\operatorname{adj}(B) \operatorname{adj}(A)$

(c) Let $M$ be an arbitrary $n \times n$ matrix. Explain why

(i) there is an $\alpha>0$ such that $M-t I$ is non-singular for $0;

(ii) the entries of $\operatorname{adj}(M-t I)$ are polynomials in $t$.

Using parts (i) and (ii), or otherwise, show that $(*)$ holds for all matrices $A, B$.

(d) The characteristic polynomial of the arbitrary $n \times n$ matrix $A$ is

$\chi_{A}(z)=(-1)^{n} z^{n}+c_{n-1} z^{n-1}+\cdots+c_{1} z+c_{0}$

By considering adj $(A-t I)$, or otherwise, show that

$\operatorname{adj}(A)=(-1)^{n-1} A^{n-1}-c_{n-1} A^{n-2}-\cdots-c_{2} A-c_{1} I .$

[You may assume the Cayley-Hamilton theorem.]