December 13, 1999
Berlekamp's comments on the Math 110 Final Exam and the solution sheet

COMMENTS ON THE Math 110 FINAL EXAM of DECEMBER 10, 1999

1: Definitions
In context, "dimension" refers to dimension of a space, not
to individual vectors.

Inverse, diagonal matrix, diagonalizable matrix, hermitian matrix,
unitary matrix, Markov matrix, and permutation matrix all should
be square.

"Determinant" is the hardest of these questions.  Defining the determinant
as the "product of the eigenvalues" might not correctly account for their
multiplicities.  Many students gave definitions that are incomplete
without also defining signs of permutation matrices.  (e.g., volume of
appropriately defined parallelpiped, or "product of pivots").  Perhaps I
could have inserted some additional terms into the
sequence of questions, preceding "determinant" with a leading
subsequence, such as "transposition matrix", "permutation matrix",
"sign of a permutation".


4C. A few students gave extremely weak (but logically "correct")
"sufficient" conditions.   The most extreme such was A = 0.
A more common variant was Ax = b.   Since least squares was such
an important topic in the early part of this course, I took off
a small number of points for answers such as these.

5. This question proved much harder than I had anticipated.  I split
the 10 points as 2 points for the 0 in the lower left corner, 1 point
each for the diagonal entries, and 6 points for the upper right entry,
with partical credit for those simple errors in its computation.

6. It was perhaps unfortunate that the pagination split this problem.
Three students wrote C backwards!  That yields a hopeless mess.
A few chose P as the inverse of my P.  With appropriate additional
changes, that works out just fine.

7DE. Some students were very confused here, despite some focus on binary
linear equations early in the course, and my writing on the blackboard
during the exam that 

	Real 1+1 = 2 > 0;  Binary 1+1 = 0



7F. A few students correctly proved that this assertion is TRUE if C and D
are nxn and n is odd.  In that case, taking determinants of both sides
of the equation

	CD = -IDC

yields (det(CD))(1 - det(-I)) = 0.  If n is odd, the latter factor is 2,
but if n is even, the latter factor is 0 and the assertion is actually
false, as seen from the counterexample on the answer sheet.
   So, for future use, I recommend an even more challenging version of
the same question:  For each positive integer n, determine whether or
not it is possible for two nonsingular real nxn matrices, C and D, to
satisfy the equation CD = -DC.
  Answer: Impossible if n is odd.  Possible if n is even, and one
construction is to put 2x2 blocks along the diagonal, with each
block equal to the 2x2 example on the answer sheet.

7G. The assertion stated not only that Lambda3 < 0 < Lambda2 <= Lambda1,
but also that Lambda2 != Lambd1.  Many omitted proof of this.
It follows from Perron's theorem, which ensures simplicity of
the largest-magnitude eigenvalue.
   For future use, consider an easier version of the same problem.
Omit the condition that the entries of the real symmetric matrix
are nonnegative, but then prove only that Lambda3 < 0 < Lambda2 <= Lambda1.

7T. Several students offered a "proof" which essentially quoted the
theorem in the book which uses this assertion as a key step in its
proof.  Such circular proofs did not get full credit!
    This underscores the point that what facts and theorems a proof
can reasonably assume is very context dependent.

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