MATH CONCEPTS: EXAM II

Skill Testing. (36 points, 3 pts each) The following problems are meant to evaluate your ability to use
mathematical concepts . Unless you’re asked to, there is no need to explain what you’re doing in this section; I’ll
mostly be looking for correct answers. In other words, don’t spend a lot of time on these! However, you do want
to show some work for partial credit… don’t just write down the answers! YOU CAN USE THE
ANSWERS FROM PREVIOUS PROBLEMS WITHOUT RE-WORKING THEM! (Just refer back to them.)

1. Determinants. Find the determinant |B| of the following matrix:

Using the definition of the determinant for a 3×3 matrix, we have:

2. Cramer’ s Rule . Use Cramer’s rule to find the value of z that will satisfy the following set of
inhomogeneous linear equations : (note that you only need to solve for z!)

In matrix notation, the above equations can be written as

Using Cramer’s rule, the value of z is given by:

Note that the value for the denominator was found in question 1.

3. Transpose. Find the transpose (A^T) of the following matrix:

4. Adjoint. Find the adjoint (A⁺) of the matrix A given in question 3.

5. Inverse. The inverse of the matrix B from question 1 is partially given by

what values of x, y, and z are need to complete the inverse matrix B^-1?

The inverse of a matrix has the following property: BB^-1 = 1, or:

To find x, we could not that, from the matrix multiply given above,

Similarly for y and z:

6. Hermitian Matrices. What values of x, y, and z will make the following matrix Hermitian?
(There may be more than one acceptable answer here.)

By inspection, x = 2–i4; y = i, and z = (any real number).

7. Eigenvalues. Find the eigenvalues (only the eigenvalues!) of the following matrix:

The eigenvalue equation would look like Ce_i = ci_ie_i. Solving this equation, we have:

Given this cubic equation , the possible values of c are c₁ = 0, c₂ = 1, and c₃ = –1.

8. Eigenvectors of a Hermitian Operator. A suave, charismatic individual comes up to you
on the street and claims that the following are both eigenvectors of the same Hermitian
operator:

Is it possible the intriguing stranger is telling the truth? Very briefly, why or why not?

No… the stranger is a dirty liar. All eigenvectors of a given Hermitian operator are
orthogonal to each other, which means that

.In this case, we have

So, the two vectors can’t possibly be eigenvectors of a particular Hermitian matrix.

9. Quadratic Equations . Solve the following equation for x:

Using the quadratic formula, we have:

10. Logarithms. Find the numerical value of the following logarithm:

(Show some work so one doesn’t get the false impression your calculator solved all of this !)

On a problem set, you proved that

which relates the logarithm of
one base to another. Using this, we can evaluate this logarithm by relating it to another, more
common, base. I’ll use the natural base, but base-10 would also work fine:

As a check, we can note that

11. Derivatives I. Evaluate the following derivative:

12. Derivatives II. Evaluate the following derivative:

(There’s no need to simplify things further like I did here… it just looks a little prettier that way.)

Concept Testing. (44 points, 11 points each.) The following problems are meant to evaluate your
understanding of mathematical ideas. Explain what you are doing… use some words! Think of it more as
writing a paper than taking an exam. Here, I’ll be looking for evidence of your mastery of the mathematical
concepts, not so much in getting the right answer. (For many of them, the “answers” are given!). You may do
these problems in any order.

13. Eigenvalues of Hermitian Operators. Use Dirac notation to prove that the eigenvalues of
a Hermitian operator are real. Be sure to explain what you’re doing, but be succinct! There’s
no need for lots of writing.

We did this one in lecture… see the notes on 9/26/08.

Update: Many people approached this proof the following way, considering the two
equations

Notice that i and j can be different here! So, we’ll have to do a little more work to show that,
for example, c_i is equal to its complex conjugate and is therefore real

Multiply the first equation on the left by and multiply the second equation on the
right by

Then, since is Hermitian, we have

Now, because the eigenvectors of Hermitian operators can be made orthonormal to each other,
we have and so we can’t just divide both sides by this value, because if i ≠ j, then

and 0/0 is undefined. Instead, we can use the Kronecker delta to say that if i ≠ j
then we can’t derive any relation between (Indeed, just knowing what one
eigenvalue is doesn’t tell us anything about what another eigenvalue might be.) However, if i
= j, then we’d have:

…and therefore the eigenvalues of a Hermitian operator must be real.

So, you could go this route, but you’d have to invoke the orthogonality of eigenvectors
of Hermitian operators, and you’d have to use the Kronecker delta appropriately!

14. Using Dirac Notation. On Exam I, you were asked to show that

Repeat that proof here using Dirac notation, only generalize it for the case where the elements
of the matrix O and vectors a and b could be complex numbers. Note that your answer will
be slightly different because you’re generalizing the proof for complex elements. [For this
problem, you can use common relations like without proving them; the primary
objective here is to assess your ability to use Dirac notation.]

In Dirac notation, the left-hand side of the above equation would be

To
introduce a basis set, we need to insert the dyadic:

Then, using the fact that we have

where we have used the fact that Continuing…

Then, since the ‘adjoint’ of a scalar is simply its complex conjugate, we have:

Note that this result is slightly different, in that we’d need to use the complex conjugate of the
elements of the vector a… this comes from generalizing the dot product to taking the adjoint of
the vector and matrix multiplying by

15. Logarithms and Inverse Trig Functions. Use what you know about complex numbers,
logarithms, and inverse trigonometry functions to prove that the logarithm of a general
complex number (z = x + iy) is given by:

[Hints: You can use the properties of logarithms here without proving them. You may find
Euler’s formula useful : Remember: partial credit is your friend! If you get
stuck, think logically as far as you can go…]

There are probably many ways one might approach this problem. I think I’ll start by
writing the general complex number with reference to plane polar coordinates :

Note that and Then, using Euler’s
equation, we have:

In this form, the original logarithm becomes:

Using the inverse tangent, we can write

And finally, with all these substitutions, we have:

Update: A lot of folks took the brute force approach to this problem, which is just as
good!… let’s see how that would have worked: first, raise both sides of the expression to a
power of e, and then use Euler’s formula:

If we focus on the first term in the square brackets , we have:

A picture might help evaluate this:

Notice that the angle θ is the result of the inverse tangent function. I.e. Using
the diagram, the cosine of this angle isSimilarly, sine of that angle is

Substituting these results in, we have:

…which is certainly true. Therefore, the original expression must have been true as well!

Important: Note that

That’s not how inverse trig functions are defined. On the left-hand side, the result is an angle.
On the right-hand side, x is an angle, and the result is the cosine of that angle divided by the
sine of that angle. To put it another way,

Instead,

Also Important: Many folks made the following algebra mistake:

if ln a = ln b + c,

The actual relation is

It’s also important not to confuse ln b + c with ln(b + c). If it was really the latter case,
parenthesis would have to be used to indicate it. However, even if it was the latter case, we’d
have:

not

16. The Product Rule. Prove the product rule of derivatives, namely:

Again: explain your thought process, but be concise!

We did this one in lecture as well… see the notes on 10/08/08.