Page 154 - Schaum's Outline of Differential Equations

CHAP. 15] MATRICES 137

The characteristic equation of A is

Hence, the eigenvalues of A are A,j = -3, X 2 = 3, and ^3 = 3. Here X = 3 is an eigenvalue of multiplicity two, while
X = -3 is an eigenvalue of multiplicity one.

15.15. Find the eigenvalues of

and

The characteristic equation of A is

which has roots A,j = 2, X 2 = -2, ^3 = -2, and X 4 = -2. Thus, X = -2 is an eigenvalue of multiplicity three, whereas
X = 2 is an eigenvalue of multiplicity one.

15.16. Verify the Cayley-Hamilton theorem for A =

2
For this matrix, we have del (A - XI) = X - 8X + 33; hence

15.17. Verify the Cayley-Hamilton theorem for the matrix of Problem 15.14.
2
For this matrix, we found del (A - XI) = -(X + 3) (X - 3) ; hence