Real symmetric matrices $\boldsymbol{A}$ and $\boldsymbol{B}$ are defined as follows:
$$\begin{aligned} & \boldsymbol{A} = \left( \begin{array}{ccc} 7 & -2 & 1 \\ -2 & 10 & -2 \\ 1 & -2 & 7 \end{array} \right), \\ & \boldsymbol{B} = \left( \begin{array}{ccc} 5 & -1 & -1 \\ -1 & 5 & -1 \\ -1 & -1 & 5 \end{array} \right). \end{aligned}$$
1. Obtain $\boldsymbol{AB}$.
Matrices $\boldsymbol{A}$ and $\boldsymbol{B}$ defined as Equations (1) and (2) satisfy $\boldsymbol{AB} = \boldsymbol{BA}$.
2. In general, two real symmetric matrices that are commutative for multiplication are simultaneously diagonalizable. Prove this for the case where all the eigenvalues are mutually different.
3. Suppose a three-dimensional real vector $\boldsymbol{v}$ whose norm is 1 is an eigenvector of $\boldsymbol{A}$ in Equation (1) corresponding to an eigenvalue $a$ as well as an eigenvector of $\boldsymbol{B}$ in Equation (2) corresponding to an eigenvalue $b$. That is, $\boldsymbol{Av} = a\boldsymbol{v}$, $\boldsymbol{Bv} = b\boldsymbol{v}$, and $\|\boldsymbol{v}\| = 1$. Obtain all the sets of $(\boldsymbol{v}, a, b)$.