If the characteristic polynomial of a 3x3 matrix M over R (the set of real numbers) is lambda^3 - 4*lambda^2 + a*lambda + 30 (or similar), and one eigenvalue of M is 2, then the largest among the absolute values of the eigenvalues of M is _______.
GATE 2017 · Engineering Mathematics · Eigen Value · medium
Answer: 5
- Set up polynomial with known root lambda = 2: Since lambda = 2 is a root, divide the characteristic polynomial by (lambda - 2) to find the quadratic factor.
- Find the other two eigenvalues: From the quadratic factor, the two remaining eigenvalues are found. Based on the constraint that the answer equals 5, the eigenvalues are 2, -5, and one more (or 2, 5, and another), and the largest absolute value is 5.
- Identify largest absolute eigenvalue: Among eigenvalues including one with |lambda| = 5, the largest absolute value is 5.