GATE | GATE-CS-2015 (Set 2) | Question 15

The larger of the two eigenvalues of the matrix

⎡ 4   5 ⎤
⎣ 2   1 ⎦

is ____
(A) 5
(B) 6
(C) 7
(D) 8

Answer: (B)
Explanation:

The character equation for given matrix is

| 4-λ    5 | = 0
| 2     1-λ|

(4-λ)*(1-λ) - 10 = 0
λ2 - 5λ - 6 = 0
(λ+1)*(λ-6) = 0
λ = -1, 6

Greater of two Eigenvalues is 6. 

Alternative way :
A short cut to this would be to remember

• The trace of a matrix is the sum of the eigen values
• The determinant is the product of the eigen values

So,

det[a] 
= 4*1 - 5*2 
= -6 

And,
tr[a] = 1+4 = 5

So,
e1*e2 = -6
e1+e2 =  5 

Hence the eigen values are -1 and 6 with 6 being the larger one.

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