342 CHAPTER 13. MATRICES AND THE INNER PRODUCT

24. Show the critical points of the function,

f (x,y,z) =−2yx2−6yx−4zx2−12zx+ y2 +2yz.

are points of the form,

(x,y,z) =(t,2t2 +6t,−t2−3t

)for t ∈ R and classify them as local minima, local maxima or saddle points.

25. Show the critical points of the function

f (x,y,z) =12

x4−4x3 +8x2−3zx2 +12zx+2y2 +4y+2+12

z2.

are (0,−1,0) ,(4,−1,0) , and (2,−1,−12) and classify them as local minima, localmaxima or saddle points.

26. Let f (x,y) = 3x4− 24x2 + 48− yx2 + 4y. Find and classify the critical points usingthe second derivative test.

27. Let f (x,y) = 3x4−5x2+2−y2x2+y2. Find and classify the critical points using thesecond derivative test.

28. Let f (x,y) = 5x4−7x2−2−3y2x2+11y2−4y4. Find and classify the critical pointsusing the second derivative test.

29. Let f (x,y,z) = −2x4− 3yx2 + 3x2 + 5x2z+ 3y2− 6y+ 3− 3zy+ 3z+ z2. Find andclassify the critical points using the second derivative test.

30. Let f (x,y,z) = 3yx2−3x2−x2z−y2 +2y−1+3zy−3z−3z2. Find and classify thecritical points using the second derivative test.

31. Let Q be orthogonal. Find the possible values of det(Q) .

32. Let U be unitary. Find the possible values of det(U) .

33. If a matrix is nonzero can it have only zero for eigenvalues?

34. A matrix A is called nilpotent if Ak = 0 for some positive integer k. Suppose A is anilpotent matrix. Show it has only 0 for an eigenvalue.

35. If A is a nonzero nilpotent matrix, show it must be defective.

36. Suppose A is a nondefective n× n matrix and its eigenvalues are all either 0 or 1.Show A2 = A. Could you say anything interesting if the eigenvalues were all either0,1,or −1? By DeMoivre’s theorem, an nth root of unity is of the form(

cos(

2kπ

n

)+ isin

(2kπ

n

))Could you generalize the sort of thing just described to get An = A? Hint: Since A isnondefective, there exists S such that S−1AS = D where D is a diagonal matrix.