The eigenvalues of A -1 are the reciprocals of theeigenvalues of a nonsingular m

Question

The eigenvalues of A-1 are the reciprocals of theeigenvalues of a nonsingular matrix A. Furthermore, theeigenvectors for A and A-1 are the same.

A = 1   2   -1
       1  0    1
       4   -4  5

I've done this problem five times, and I haven't gotten reciprocaleigenvalues. Given that I must be doing something wrong, couldsomeone please walk me through this problem?

Thanks

Explanation / Answer

To find eigen values of a matrix A det(A-xI)=0 Characteristic polynomial:     x^3 - 6x^2 + 11x - 6 Real eigenvalues:     {1, 2, 3} Eigenvector of eigenvalue X1 = 1:     E1 = (-1, 1, 2) Eigenvector of eigenvalue X2= 2:     E2= (-2, 1, 4) Eigenvector of eigenvalue X3= 3:     E3 = (-1, 1, 4) Now as given in the question : eigen vectors are same for A andA-1 => [A-1-K1I]*E1=0   , where K1 is anyeigen value of A-1 Now multiplying the above equation with A we get                         A*[A-1-K1I]*E1 = 0                         [I-K1A]*E1=0    This above equation onrearranging gives                                                 (-K1)[A-(1/K1)I]E1=0                          => [A-(1/K1)I]E1 =0                           => X1=1/K1 Therefore it is prooved that eigen values of matrix A-1are reciprocal with eigen values of matrix A

Related Questions

Navigate

• Browse (All)
• Browse T
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.