Gate 2015 CE set 2

Question

The two eigen values of the matrix $\begin{bmatrix} 2 & 1\\ 1& p \end{bmatrix}$ have a ratio of 3:1 for p= 2.

What is another value of p for which eigenvalues have the same ratio of 3:1?

A)-2   b) 1  c) 7/3   d)14/3

Answer 1

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