Symmetric matrix with $a_{ij} = 0$ for all $|i - j| > 1$ has all eigenvalues of multiplicity $1$ 
1 . Let $A = (a_{ij})$ be a real $n \times n$ matrix such that $a_{ij} = a_{ji}$ for all $1 \leq i,j \leq n$ and $a_{ij} = 0$ for $|i-j|>1$. Moreover $a_{ij}$ is non-zero for all $i$,$j$ satisfying $|i-j| = 1$. Show that all the eigenvalues of $A$ are of multiplicity $1$.
2 . Give examples of 2 real $n \times n$ matrices  $X = (x_{ij})$, $Y = (y_{ij})$ where $x_{ij} = x_{ji}$ and $y_{ij} = y_{ji}$ for all $1 \leq i,j \leq n$ so that $xX +yY$ has $n$ non-repeated eigenvalues for all real numbers $x$, $y$ where $x$, $y$ are not zero simultaneously.

Thank you for any help.
 A: 2) Let $X,Y$ be linear independent real symmetric matrices of order 2 and trace zero.
Let $Z$ be any linear combination of X and Y. Notice that $Z$ has the same properties. Therefore its eigenvalues $(Z$ is diagonalizable since it is real symmetric$)$ have oppositive signs $($their sum must be zero$)$, unless $Z$ is the zero matrix.
But it occurs only if $Z$ is the trivial combination of $X$ and $Y$. 
Now for matrices of order $2k$ $(2k+1)$, instead of $X$ and $Y$, use $F(X)$ and $F(Y)$ $(G(X)$ and $G(Y))$.
$F(X)=\left(\begin{array}{cccc}
X & 0 & \dots & 0 \\ 
0 & 2X & \dots & 0 \\ 
\vdots & \vdots & \ddots & 0 \\ 
0 & 0 & 0 & kX
\end{array} \right)_{2k\times2k}$ $G(X)=\left(\begin{array}{cc}
F(X) & 0_{2k\times 1} \\ 
0_{1\times 2k} &  0_{1\times 1}
\end{array} \right)_{2k+1\times 2k+1}$
If $Z$ is any linear combination of $X$ and $Y$ then $F(Z)$ $(G(Z))$ is be the respective linear combination of $F(X)$ and $F(Y)$ $(G(X)$ and $G(Y))$. 
If $a,-a$ are the eigenvalues of $Z$ then $a,-a,2a,-2a,\dots,ka,-ka$ are the eigenvalues of $F(Z)$ $(a,-a,2a,-2a,\dots,ka,-ka,0$ are the eingevalues of $G(Z))$.
A: Hints.


*

*Let $\lambda$ be an eigenvalue of $A$. As $A$ is diagonalisable, can you relate the geometric multiplicity of $\lambda$ to the rank of $\lambda I-A$? Now, let $B$ be the submatrix obtained by deleting the first row and last column of $\lambda I-A$. What is the rank of $B$? Then, what is the rank of $\lambda I-A$?

*Split a matrix in the form of $A$ in part 1 into two appropriate symmetric matrices!

