Let the monic polynomials $p_{n}, n \geqslant 0$, be orthogonal with respect to the weight function $w(x)>0, a<x<b$, where the degree of each $p_{n}$ is exactly $n$.

(a) Prove that each $p_{n}, n \geqslant 1$, has $n$ distinct zeros in the interval $(a, b)$.

(b) Suppose that the $p_{n}$ satisfy the three-term recurrence relation

$$p_{n}(x)=\left(x-a_{n}\right) p_{n-1}(x)-b_{n}^{2} p_{n-2}(x), \quad n \geqslant 2$$

where $p_{0}(x) \equiv 1, p_{1}(x)=x-a_{1}$. Set

$$A_{n}=\left(\begin{array}{ccccc} a_{1} & b_{2} & 0 & \cdots & 0 \\ b_{2} & a_{2} & b_{3} & \ddots & \vdots \\ 0 & \ddots & \ddots & \ddots & 0 \\ \vdots & \ddots & b_{n-1} & a_{n-1} & b_{n} \\ 0 & \cdots & 0 & b_{n} & a_{n} \end{array}\right), \quad n \geqslant 2 .$$

Prove that $p_{n}(x)=\operatorname{det}\left(x I-A_{n}\right), n \geqslant 2$, and deduce that all the eigenvalues of $A_{n}$ reside in $(a, b)$.