Final Exam, Math. 221 (s. 02), 11 May 2004

Scoring: Each of questions 1-10 is worth twelve points. Partial credit may be earned.
Top score possible is 120 points, with 100 points probably representing a very respectable grade.
In any event, letter grades will be distributed along a suitable "curve."

Suggested answers starting to be available now; more details soon.

1. For which value(s) of the constant c does the system of equations

  x + y + 2z = a
  x         + z = b
2x + y + 3z = c

have

(i) infinitely many solutions?
(ii) more than one solution, but not infinitely many?
(iii) exactly one solution?
(iv) no solution at all?

Explain your reasoning, and exhibit the solutions that are available.

2. Solve the system of equations

  3/x + 2/y − 1/z = −15
  5/x + 3/y + 2/z = 0
  3/x + 1/y + 3/z = 11

for x, y, and z, explicitly explaining all relevant strategies and calculations.

3. Defend or rebut the following allegation(s).

(i) For all n×n matrices A and B, Trace(A +/− B) = Trace(A) +/− Trace(B).
(ii) For all n×n matrices A and B, Trace(AB) = Trace(A)Trace(B).
(iii) For all n×n matrices A and B, Trace(AB) = Trace(BA).

[Suggestion: for item (iii), what's the relevance of the matrix C whose (ij)^th entry c_ij
is the product c_ij = a_ij b_ji of the (ij)^th entry a_ij of A with the (ij)^th entry b_ji of B^T?
And can you add up all the entries of C in different ways, still obtaining the same total sum?]

4. Suppose a 5×5 matrix M has one of the forms

    | * * 0 0 0 |       | * 0 0 0 * |       | 1 * * * * |
    | * * 0 0 0 |       | 0 0 * 0 0 |       | 0 1 * * * |
A = | * * 0 0 0 | , B = | 0 * 0 * 0 | , C = | 0 0 * * * | .
    | * * * * * |       | 0 0 * 0 0 |       | 0 0 0 1 * |
    | * * * * * |       | * 0 0 0 * |       | 0 0 0 0 1 |

(i) What are all the values can you obtain for det(M) (M = A, B, C) by substituting
numerical values (not necessarily all the same) for the *'s? Explain your reasoning.
(ii) Which of these forms can be invertible, and when exactly?

5. What can one say about the behavior of the rows of a square matrix A whose columns are all mutually perpendicular and of length one? Explain your reasoning.

6. A (new) definition: a (square) matrix M (n×n, say, with n > 1) is said to be idempotent if MM = M.

(i) Show that, if a real number λ is an eigenvalue of an idempotent matrix, then either λ = 0 or λ = 1.
(ii) If square matrix A is idempotent, and I is the identity matrix the same size and shape as A,
show that I − A is idempotent too.
(iii) Show that when idempotent A is not the zero matrix, then 1 is an eigenvalue for A,
while when idempotent A is not the identity matrix, then 0 is an eigenvalue for A.
(iv) Given column vector z in Rⁿ, let x = Az and y = (I − A)z.
Evaluate Ax, Ay, (I − A)x, (I − A)y, and x + y, and determine whether or not x and y are linearly independent.
(v) Discuss any relationships you can now discern among
the column spaces, the eigenspaces, and the nullspaces of A and of I − A.
(vi) Establish why every idempotent matrix must be diagonalizable; or, if that's not true, present a counterexample.

7. Suppose n > 1 and the n×n matrix M is nilpotent (another new definition: we call
a (square) matrix M nilpotent if some iterated m-fold product of M with itself vanishes:

(†) M^m = M · M · ... · M = 0   (here 0 stands for the matrix with all zero entries)).
                 ← m factors →

(i) Determine the possible eigenvalues of the nilpotent matrix M.
(ii) With k > 0 an arbitrary integer, how do the column spaces of M^k and M^k+1 compare?
(iii) Explain why, if ever M^k = M^k+1, it will follow that M^k = 0.
(iv) Show that, for n×n nilpotent M, an integer m as in (†) can always be found satisfying m < n.

8. Consider the following four vectors in R₃:
the vector from the origin to (1, 1, −1), that from the origin to (1, −1, 1),
that from the origin to (−1, 1, 1), and that from the origin to (−1, −1, −1).

(i) What are their lengths? (Needed for part (ii).)
(ii) What are the angles (or, at least, the cosines of the angles) between any two of them?
(NB: there are 6 such angles(!). The four non-(0, 0, 0) points mentioned form the vertices of a tetrahedron.)

9. By a zero vector for a vector space V (with addition +) some people mean any vector z satisfying:

for all vectors v in V, z + v = v = v + z.

By evaluating z + z' two different ways, show that any two zero vectors z and z' for V must be equal.

10. Describe briefly any two components of this course that you feel this exam has unfairly ignored, despite their being absolutely central to the development of more advanced vector/matrix ideas.

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