please answer Consider the matrix A 2 2 1 0 1 6 0 2 2 Find

please answer

Consider the matrix A = [2 2 1 0 -1 6 0 2 -2] Find the eigenvalues of A and bases for the corresponding eigenspaces. Determine the geometric and algebraic multiplicities of each eigenvalue and whether A is defective or not.

Solution

Trace of the matrix = 2 - 1 - 2 = -1

determinant of the matrix = -20

The characteristic equation of matrix is,

^{3 2} + 16× 20 = (1)( 2)( 2)( + 5)

1 = 2

2 = 2

3 = -5

For 1 = 2 and 2 = 2:

Eigen vector = column matrix of (x1 0 0)

3 = -5 ======> Eigen vector = column matrix of (2/7*x3 -1.5*x3 x3)