What is the adjugate adj $(A)$ of an $n \times n$ matrix $A$ ? How is it related to $\operatorname{det}(A) ?$

(a) Define matrices $B_{0}, B_{1}, \ldots, B_{n-1}$ by

$$\operatorname{adj}(t I-A)=\sum_{i=0}^{n-1} B_{i} t^{n-1-i}$$

and scalars $c_{0}, c_{1}, \ldots, c_{n}$ by

$$\operatorname{det}(t I-A)=\sum_{j=0}^{n} c_{j} t^{n-j}$$

Find a recursion for the matrices $B_{i}$ in terms of $A$ and the $c_{j}$ 's.

(b) By considering the partial derivatives of the multivariable polynomial

$$p\left(t_{1}, t_{2}, \ldots, t_{n}\right)=\operatorname{det}\left(\left(\begin{array}{cccc} t_{1} & 0 & \cdots & 0 \\ 0 & t_{2} & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & t_{n} \end{array}\right)-A\right)$$

show that

$$\frac{d}{d t}(\operatorname{det}(t I-A))=\operatorname{Tr}(\operatorname{adj}(t I-A))$$

(c) Hence show that the $c_{j}$ 's may be expressed in terms of $\operatorname{Tr}(A), \operatorname{Tr}\left(A^{2}\right), \ldots, \operatorname{Tr}\left(A^{n}\right)$.