13.9. EXERCISES 327

24. Fill in the missing entries to make the matrix orthogonal.−1√

2−1√

61√3

1√2

√6

3

 .

25. Fill in the missing entries to make the matrix orthogonal.23

√2

216

√2

23

0

26. Fill in the missing entries to make the matrix orthogonal.

13 − 2√

523 0

415

√5

27. Find the eigenvalues and an orthonormal basis of eigenvectors for A. Diagonalize A

by finding an orthogonal matrix U and a diagonal matrix D such that UT AU = D.

A =

 −1 1 11 −1 11 1 −1

 .

Hint: One eigenvalue is -2.

28. Find the eigenvalues and an orthonormal basis of eigenvectors for A. Diagonalize Aby finding an orthogonal matrix U and a diagonal matrix D such that UT AU = D.

A =

 17 −7 −4−7 17 −4−4 −4 14

 .

Hint: Two eigenvalues are 18 and 24.

29. Find the eigenvalues and an orthonormal basis of eigenvectors for A. Diagonalize Aby finding an orthogonal matrix U and a diagonal matrix D such that UT AU = D.

A =

 13 1 41 13 44 4 10

 .

Hint: Two eigenvalues are 12 and 18.