The characteristic polynomial of a 5 x5 matrix is given belo

The characteristic polynomial of a 5 x5 matrix is given below. Find the eigenvalues and their multiplicities. 3) A) 0 (multiplicity 1), 9 (multiplicity 3),-5 (multiplicity 1) B) 0 (multiplicity 2),-9 (multiplicity 2), 5 (multiplicity 1) C) 0 (multiplicity 2), 9 (multiplicity 2),-5 (multiplicity 1) D) 0 (multiplicity 2),-9 (multiplicity 2),-5 (multiplicity 1) Diagonalize the matrix A, if possible. That is, find an invertible matrix P and a diagonal matrix D such that A PDP-1 4) A 0 -0 5-18 4) A) B) 1 0-1 1-9-1 P=10-4 01,D- P= -9-4 01.D-10-40 0 0 -3 01 0 0 0-3 4 0 0 -4 3 C) D) P-94 0D00 0 0 3

Solution

3) x^5-13x^4-9x^3+405x^2= x^2(x+5)(x-9)^2. So (-5) is a simple root, 0 and 9 are double root for the given polynomial.Hence the right option is (C)

4) Since the eigenvalues of A (root of the characteristic polynomial of the matrix A) are -4, (-7+sqrt(61))/2, and

(-7-sqrt(61))/2, none of the given options are correct.