GATE | GATE-CS-2015 (Set 1) | Question 65

Consider the following 2 × 2 matrix A where two elements are unknown and are marked by a and b. The eigenvalues of this matrix are –1 and 7. What are the values of a and b?

A = | 1  4 |
    | b  a |

(A) a = 6, b = 4
(B) a = 4, b = 6
(C) a = 3, b = 5
(D) a = 5, b = 3


Answer: (D)

Explanation:

The character equation for given matrix is

| 1-λ    4 | = 0
|  b    a-λ|

(1-λ)*(a-λ) - 4b = 0

Putting λ = -1,
=> (1 - (-1)) * (a - (-1)) - 4b = 0
=> 2a + 2 - 4b = 0
=> 2b - a = 1


Putting λ = 7,
=> (1 - 7) * (a - 7) - 4b = 0
=> -6a + 42 - 4b = 0
=> 2b + 3a = 21

Solving the above two equations, we get
a = 5, b = 3


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