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suparna kar
asked
in Linear Algebra
Aug 18, 2018

1,935 views
6 votes

Best answer

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