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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
in Linear Algebra by Junior (741 points) | 191 views

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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

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