19.9. EXERCISES 427

9. Given a quadratic form in three variables, x,y, and z, show there exists an orthogonalmatrix U and variables x′,y′,z′ such that

(x y z

)T= U

(x′ y′ z′

)T withthe property that in terms of the new variables, the quadratic form is

λ 1(x′)2

+λ 2(y′)2

+λ 3(z′)2

where the numbers, λ 1,λ 2, and λ 3 are the eigenvalues of the matrix A in Problem 8.

10. If A is a symmetric invertible matrix, is it always the case that A−1 must be symmetricalso? How about Ak for k a positive integer? Explain.

11. If A,B are symmetric matrices, does it follow that AB is also symmetric?

12. Suppose A,B are symmetric and AB = BA. Does it follow that AB is symmetric?

13. Here are some matrices. What can you say about the eigenvalues of these matricesjust by looking at them?

(a)

 0 0 00 0 −10 1 0

(b)

 1 2 −32 1 4−3 4 7

(c)

 0 −2 −32 0 −43 4 0

(d)

 1 2 30 2 30 0 2



14. Find the eigenvalues and eigenvectors of the matrix

 c 0 00 0 −b0 b 0

 . Here b,c are

real numbers.

15. Find the eigenvalues and eigenvectors of the matrix

 c 0 00 a −b0 b a

. Here a,b,c

are real numbers.

16. Find the eigenvalues and an orthonormal basis of eigenvectors for A.

A =

 11 −1 −4−1 11 −4−4 −4 14

 .

Hint: Two eigenvalues are 12 and 18.

17. Find the eigenvalues and an orthonormal basis of eigenvectors for A.

A =

 4 1 −21 4 −2−2 −2 7

 .

Hint: One eigenvalue is 3.