Let A 2 0 0 3 2 0 6 1 1 Find the characteristic polynomial

Let A = [2 0 0 3 2 0 6 -1 1] Find the characteristic polynomial of A. Find the eigenvalues and bases for the corresponding eigenspaces of A. State the algebraic multiplicity and geometric multiplicity of each eigenvalue of A. Is A is diagonalizable? Justify your answer. Compute A^7 v vector, where v vector = [-17 3 3].

Solution

a) Characterstic equation :

(2-x)(2-x)(1-x) -0 + 6(0) =0

-x^3 +5x^2 - 8x +4 =0

b) Eigen values

x = 1 ,2 , 2

Eigen vectors : x= 2; ( 1,0,0)

x = 1 ; (-9,1,1)

c) x = 1 algebraic multiplicty = 1 , geometric multiplicty = 1

x = 2 ; algebraic multiplicty = 2 , geometric multiplicty = 2

d) Matrix is not diagonlisable as no. of eigen vectors and dimension of original matrix are not same