18 A is a 4 by 4 matrix with characteristic polynomial 1 ga

18. A is a 4 by 4 matrix with characteristic polynomial (1 - gamma)(3 - 5 gamma + gamma^2)(2 - gamma). Give the smallest eigenvalue of the matrix A + 4I.

Solution

eigen values of A+4*I will be lambda+4 where lambda is the eigen value for A.

now from characterisitics polynomial,

eigen values of A are:

lambda=1
lambda=2
lambda=4.3
lambda=0.69

so smallest eigen value=0.69

hence smallest eigen value for A+4I will be 4.69