Determinants

One of the properties of any matrix is its determinant. For a 3×3 matrix, its
de terminant is defined as follows:

For remembering how to find the determinant of a 3×3 matrix, the following mental
image is helpful:

Why would anyone want to make this definition? Well, it turns out that
determinants have several interesting properties, which are used in the study of
molecular quantum mechanics. Originally, these properties were discovered as
mathematicians studied simultaneous Linear Equations (which we’ll do in a bit as well).
These interesting properties are as follows: (I’ll give you a handout with these.)

1. If all the elements of any row or column in a matrix are all zero , its determinant is
zero. For example:

2. If any two rows or columns in a matrix are interchanged, the determinant
changes sign. For example:

3. If two rows or columns of a matrix are the same, then the determinant for that
matrix is zero. For example:

4. If each element in any row or column in a determinant is multiplied by the same
number c , the value of the determinant is multiplied by c For example:

5. If each element in any row or column is multiplied by a constant and added to
each corresponding element in another row or column, the determinant is
unchanged. For example:

As an exercise, prove all of these properties for a general 3×3 determinant by
writing out the determinant explicitly.

H. Solution for a System of Inhomogeneous Linear Equations (§10.2)

All this matrix stuff has all seemed a bit abstract, I’m sure. (A bit??!?) Let’s try
and see one of the very useful applications of matrix algebra in this section . Consider
the following three equations involving three unknowns, x₁, x₂, and x₃:

One could solve these equations for x₁, x₂, and x₃ using the usual method; solve one
equation for x₁, plug that result into the second, etc., etc. (You’ve no doubt solved for
two equations and two unknowns before… this would be the same, only more steps .)
Such a method is tedious. Instead, this system can be solved using Cramer’s rule. Note
that the above equations could be written using matrix and vector notation. Recall a
previous example :

If we let

Then we recover the original three equations we started with. Now, consider the
determinant of the matrix R:

According to property 4 of determinants, we can write

Then, using property 5 of determinants (twice!), we can further write

Note that the first column is simply the left -hand side of the three equations we started
with:

Rearranging this, we have

Similar manipulations will give us expressions for the other unknowns:

Notice that in the numerator, the column corresponding to the x to be found has been
replaced by the inhomogeneity (the RHS of each of the three equations). If you evaluate
these determinants, you will find that

…and sure enough, these are the solutions to our original system of equations:

This technique for solving a system of linear equations is called Cramer’s Rule.

Note that if |R| = 0, then the solutions for this set of inhomogeneous linear
equations will go to infinity according to Cramer’s rule. In fact, there may still be a
solution, but the fact that |R| = 0 means there is a linear dependence in the set of equations.
In other words, the three equations are not unique, and you don’t have three equations
for three unknowns. (This is ultimately a consequence of property 3 of determinants,
but it is often coupled with properties 4 and 5. See for example Problem 10.6 on p. 316
of Mortimer.) If the determinant of a matrix is zero, that matrix is called singular.

I. Solution for a System of Homogeneous Linear Equations (§10.2)

Homogeneous linear equations are a set of equations that all equal zero:

Written in matrix form, we have:

Ax = 0

Where 0 is a vector where all the components are zero. Using Cramer’s rule, we would
have the following solutions for the components of x:

Where we have used property 1 of determinants. If |A| ≠ 0, then we have x₁ = x₂ = x₃ = 0:
a trivial solution. The only way there can be a non-trivial solution is when |A| = 0.
Then, 0/0 is undefined, but it could be a finite answer. (Of course, the drawback is that
we can ’t use Cramer’s rule to find the answer !) When |A| = 0, A is called a singular
matrix. By extension with the previous section, this also means the equations must
have a linear dependence in order to find a solution. Otherwise, the only solution is the
trivial one (i.e. x = 0).

To summarize: Given the set of homogeneous linear equations

Ax = 0

There can only be a non-trivial answer if A is singular. We will use this idea again in a
moment when we consider a special type of homogeneous equations called eigenvalue
equations, and see how to solve such a system of equations.

J. The Cross Product Revisited (Problem Set)

One last note related to determinants. If you recall our previous work on cross
products, it turns out that the determinant provides a very nice way to redefine that
concept. Given two general vectors a and b in the Cartesian basis, their cross product is
defined as

Note that the unit vectors for the basis set being used for the vectors a and b are the first
row of the matrix, the components for the first vector are the second, and the
components for the second vector are the third. To see the equivalency of this
definition, consider using it on the cross product of two unit vectors:

You might use this technique to verify the other results for cross products of unit
vectors derived earlier. (If you’re like me and don’t really care for the right-hand rule,
this alternate definition is a life saver!)

K. Matrix Terminology

The following definitions and termino logy are commonly encountered when
working with matrices. As we consider them, we’ll also be able to generalize what
we’ve done to complex vectors and matrices.

1. Transpose

The transpose (A^T) of a matrix A is defined as follows:

For example:

I.e. The horizontal rows and vertical columns have been swapped. Note that the
diagonal elements of the matrix (in this case, the 1, 5, and 9) are unchanged.

Using this along with the general definition of a matrix multiply throws new
light on the dot product of two vectors. If we think of a column vector as a 1×3 matrix:

then its transpose of b is given by

If you consider a to be a 3×1 matrix and a^T to be a 1×3 matrix, then the matrix
product of these two matrices is:

Note that this result is a number (of, if you like, the “matrix” only has one element), and
that number is the same result you’d get from taking the dot product a * a. Therefore,
you will often see the dot product of vectors written as a^T a, a matrix multiplication.
Note that, using the Pythagorean theorem, the length of the vector a is given by

As mentioned previously, for a one-dimensional vector (i.e. a scalar), this “length”
corresponds to the absolute value, which is why we use the same symbol (two vertical
lines) here.

Use this idea to show that a b^T is a matrix whose elements are given by

If you’re having trouble seeing the above results, you might look at the problem
a bit more formally and use our general formula for matrix multiplication. Let

Written this way, it may be easier to see that a really is a 3×1 matrix. Similarly, let

Where we’ve used the definition of the transpose to generate A₁₁, A₁₂, and A₁₃. Using the
general formula for matrix multiplication, we have:

However, since a^T has only one row, i must be 1. Also, since a has only one column, j
must be 1. Therefore, we really have:

Again, the dot product! Similarly, if we were to consider

But this time, k can only be equal to 1. Since k can only have one value, the sum over k
has no meaning, and we’re left with

So, for example, the first element of the resulting matrix is:

…as we saw previously.