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 3k2 = 2p-1
p= 4k-2..put this in 3k2 = 2p-1
3k2= 2(4k-2)-1 => 3k2= 8k-4-1 => 3k2-8k+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