Characteristics equation :-
(2-⋋)(p-⋋) -1 = 0
By solving :
⋋^{2} -(2+p)⋋+(2p-1) =0
Shortcut :- ⋋^{2} - (Trace of matrix)⋋ + (Det. of matrix) =0
Now let ⋋_{1 }and ⋋_{2} be two eigen values hence the roots of this equation.
So from quadratic equation properties,
Sum of roots = -b/a and Product of roots= c/a
⋋_{1} + ⋋_{2} =(2+p) ... (1)
⋋_{1}*⋋_{2= }(2p-1) ... (2)
⋋_{1}:⋋_{2} = 3:1 (given)
So let ⋋_{2}=k then ⋋_{1}=3k where k is a constant
Put them in equation 1 and 2
4k= 2+p and 3k^{2} = 2p-1
p= 4k-2..put this in 3k^{2} = 2p-1
3k^{2}= 2(4k-2)-1 => 3k^{2}= 8k-4-1 => 3k^{2}-8k+5 = 0
After solving this we get k=5/3 or 1.
When k=5/3, p=4*(5/3)-2 = 14/3
when k=1, p=4*(1)-2 =2 (already given)
So p=14/3