1. The matrix A given below has two distinct integer eigenvalues. Compute and factor the determinant of A xI, where I is the 4 4 identity matrix.

A = 38 44 104 176

9 2 36 56

8 8 20 32

3 2 12 20 .

2. For each eigenvalue of the matrix A in Problem 1, find a basis for the corresponding eigenspace.

3. (a) Find three orthonormal vectors u1, u2, and u3 in R 3 such that the solution set of the equation 6xy + 8xz + 6yz = 1 can be expressed in the form

S = {Xu1 + Y u2 + Zu3 : 1X 2 + 2Y 2 + 3Z 2 = 1},

where 1, 2, and 3 are the eigenvalues of the associated real symmetric matrix A.

(b) Use the information in part (a) to graph the equation, superimposing the vectors u1, u2, and u3.