Find the characteristic polynomial and the eigenvalues of th

Find the characteristic polynomial and the eigenvalues of the following matrices and if the eigenvalues are integers, find the eigenvectors. (a) A = [1 0 3 2] (b) A = [5 2 3 4] (c) A = [1 2 2 1] (d) A = [3 5 2 1]

Solution

7(a). The characteristic polynomial of A is det(A- I₂) = (1- )(2- )-3*0 = (-1)( -2) = ²-3 +2. The eigenvalues of A are ₁ = 2 and ₂ = 1.   The eigenvector of A corresponding to the eigenvalue 2 is solution to the equation (A-2I₂)X = 0. To solve this equation, we reduce A-2I₂ to its RREF. The RREF is

1

-3

0

0

Thus, if X = (x,y)^T, then the equation (A-2I₂)X = 0 is equivalent to x-3y = 0 or, x = 3y. Then X = (3y,y)^T = y (3,1)^T. Hence, the eigenvector of A corresponding to the eigenvalue 2 is (3,1)^T. Similarly, the eigenvector of A corresponding to the eigenvalue 1 is (1,0)^T.

(b) The characteristic polynomial of A is det(A- I₂) =(5- )(4- )-3*2 = ²-9 +14 = (-2)( -7). The eigenvalues of A are ₁ = 7 and ₂ = 2. The eigenvectors of A corresponding to the eigenvalues 7 and 2 are solutions to the equation (A-7I₂)X = 0 and (A-2I₂)X = 0 respectively. The RREF of A-7I₂ is

1

-3/2

0

0

Thus, if X = (x,y)^T, then the equation (A-7I₂)X = 0 is equivalent to x-3y/2 = 0 or, x = 3y/2. Then X = (3y/2,y)^T = 2y(3,2)^T. Hence, the eigenvector of A corresponding to the eigenvalue 7 is (3,2)^T. Similarly, the eigenvector of A corresponding to the eigenvalue 2 is (-1,1)^T.

(c ) The characteristic polynomial of A is det(A- I₂) =(1- )(1- )-2*2 or, ²-2 -3 or, (+3)( -1). The eigenvalues of A are ₁ = 3 and ₂ = -1.   The eigenvector of A corresponding to the eigenvalues 3 and -1 are solutions to the equation (A-3I₂)X = 0 and (A+I₂)X = 0. The RREF of A+I₂ is

1

1

0

0

Thus, if X = (x,y)^T, then the equation (A+I₂)X = 0 is equivalent to x+y = 0 or, x = -y. Then X = (-y,y)^T =            -y(-1,1)^T. Hence, the eigenvector of A corresponding to the eigenvalue -1 is (-1,1)^T. Similarly, the eigenvector of A corresponding to the eigenvalue 3 is (1,1)^T.

(d) The characteristic polynomial of A is det(A- I₂) =(3- )(1- )-2*5 = ²-4 -7= [-(2+11)][ -(2-11 )]. The eigenvalues of A are ₁ = 2+11 and ₂ = 2-11.

      

1	-3
0	0