We know that MX= ⋋X where X is the eigen vector correspondig to eigen value ⋋
So for ⋋=1,
$\begin{pmatrix} a &b &c \\ b &d &e \\ c &e &f \end{pmatrix} * \begin{bmatrix} 1\\ 1 \\ 1 \end{bmatrix} = 1* \begin{bmatrix} 1\\ 1 \\ 1 \end{bmatrix}$
a+b+c=1 --> (1)
b+d+e=1 --> (2)
c+e+f=1 --> (3)
For ⋋=0
$\begin{pmatrix} a &b &c \\ b &d &e \\ c &e &f \end{pmatrix} * \begin{bmatrix} 1\\ -1 \\ 0 \end{bmatrix} = 0* \begin{bmatrix} 1\\ -1 \\ 0 \end{bmatrix}$
a-b=0 => a=b
b-d=0 => b=d
c-e=0 =>c=e
So, a=b=d and c=e
Let us say a=b=d=x and c=e=y
Now the matrix becomes :
$\begin{bmatrix} x & x &y \\ x & x & y\\ y &y & f \end{bmatrix}$
Given eigen values are 1,0,3
Trace of matrix = sum of eigen values
x+x+f=1+0+3 = 4
=>2x+f=4 --> (4)
Also from equ (1)
x+x+y=1 => 2x+y=1 --> (5)
From equ (2)
y+y+f=1 => 2y+f=1 --> (6)
From (4), (5) and (6),
f=1-2y (from (6) )
Putting this in (4)
2x+f=4 => 2x+1-2y=4
=> 2x-2y =3
2x+y=1 (from (5) )
Solving these two we get y= -2/3
f=1-2y (from (6) )
= 1 - 2(-2/3) = 1 +4/3 = 7/3
3f = 7.