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