The trace and determinant of a 2 x 2 matrix are known to be -2 and -35...
Given: Trace = -2, Determinant = -35, Eigenvalues = -30, -5b
Finding Eigenvalues:
We know that the sum of eigenvalues is equal to the trace of the matrix, and the product of eigenvalues is equal to the determinant of the matrix.
Let λ1 and λ2 be the eigenvalues of the matrix.
λ1 + λ2 = Trace = -2
λ1 * λ2 = Determinant = -35
Substituting λ1 = -30 in the first equation, we get:
-30 + λ2 = -2
λ2 = -2 + 30 = 28
Substituting λ1 = -5b in the first equation, we get:
-5b + λ2 = -2
λ2 = -2 + 5b
Substituting the values of λ2 in the second equation, we get:
-30 * 28 = (-5b) * (-2 + 5b)
-840 = 10b^2 - 5b
2b^2 - b + 168 = 0
Solving this quadratic equation, we get:
b = -7 or b = 12
Therefore, the eigenvalues are -30 and -35.
Hence, option C is correct: -7 and 5
The trace and determinant of a 2 x 2 matrix are known to be -2 and -35...