Let $\left\{P_{n}\right\}_{n=0}^{\infty}$ be the sequence of monic polynomials of degree $n$ orthogonal on the interval $[-1,1]$ with respect to the weight function $w$.

Prove that each $P_{n}$ has $n$ distinct zeros in the interval $(-1,1)$.

Let $P_{0}(x)=1, P_{1}(x)=x-a_{1}$, and let $P_{n}$ satisfy the following three-term recurrence relation:

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

Set

$$A_{n}=\left[\begin{array}{ccccc} a_{1} & b_{2} & 0 & \cdots & 0 \\ b_{2} & a_{2} & b_{3} & \ddots & \vdots \\ 0 & \ddots & \ddots & \ddots & 0 \\ \vdots & \ddots & b_{n-1} & a_{n-1} & b_{n} \\ 0 & \cdots & 0 & b_{n} & a_{n} \end{array}\right]$$

Prove that $P_{n}(x)=\operatorname{det}\left(x I-A_{n}\right), n \geqslant 1$, and deduce that all the eigenvalues of $A_{n}$ are distinct and reside in $(-1,1)$.