Determinants
Problems
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Determinants
Determinants
are important in many different kinds of analysis. The most common
way you can encounter determinants is when solving simultaneous
systems of linear equations. Many analysis programs have
the ability to do determinant computations. However, it can be important
to understand how to use determinants, and there may be occasions when
you want to be able use calculate a small determinant. In this lesson
we will review how determinants arise and how small sets of simultaneous
equations manually.
We start with a simple example. Assume that you have two simultaneous
linear equations in two unknowns, x and y. Here are the equations:
a*x + b*y = e
c*x + d* y = f
You may know a number of ways to solve these
equations for x and y. However, here we are going to present the
solution method using determinants. The solution is easily stated:
Similarly, the solution for y is given as:
Here, N_{1},
N_{2}
and D are all determinants. We
will look at the denominator determinant,
D,
first since it appears in the solution for every variable. It is
defined as:
We say that "D
is the determinant of the system of equations". D
is computed from the coefficients of the variables (x and y) on the left
hand side of the set of equations. We can view this determinant as
the determinant of a matrix formed from those coefficients. This
determinant is actually computed as:
D = ad  cb
Paraphrasing this result
we see that D is the product of the two elements on the descending
diagonal (that is "a" times "d") minus the product of the two elements
on the ascending diagonal (that is "c" times "d").
The other two determinants, N_{1}
and N_{2}, are computed in
the same way, but not with the same coefficients. N_{1}
is similar to D but for N_{2}
the first column is replaced by the values on the right hand side of the
set of equations. This means:
So, we find:
N_{1}
= ed  fb
which is computed in exactly the same manner
as D was computed.
In any event, we can note the following:

We have the solution
for x:

We can also compute the
solution for y.
However, in the set of equations y appears in the second column, so to
compute N_{2} we replace
the second column in D, and then compute
the determinant.

The characteristic
determinant, D, appears in the denominator
of the solution for every variable.

The method can be a numerical
method or a symbolic method. In other words, you can use the technique
with numerical coefficients, or with symbolic coefficients.
Next,
we need to consider what happens when there are more than two simultaneous
equations. We start in that direction by considering a set of three
simultaneous equations.
a*x + b*y + c*z = j
d*x + e*y + f*z = k
g*x + h* y + i*z =
l
Here, a, b, c ....i, are known coefficients
(either numerical or symbolic). And, x, y and z are the unknown variables
that we want to find. The values, j, k and l are known values.
In a circuit, j, k and l would be the parts of the equation that contain
independent voltage and current sources.
The solution in this case is expressed in exactly the same way as it was
for the system of two simultaneous equations:
However, in this case the determinants are determinants
of 3 x 3 matrices  with 9 elements. The characteristic determinant,
D, looks like this:
You can see that there are now nine elements
in the determinant. The question is "How is
the determinant calculated?". In this case, the computation
is not the
product of the two elements on the descending diagonal minus the product
of the two elements on the ascending diagonal.
There are six terms in the expression for the determinant.:
D = aei + bfg +
cdh  gec ahf  dbi
The first term is the product of the three elements on the descending diagonal.
And, the three elements on the ascending diagonal are also multiplied together
and they show up with a negative sign. All of that is the same as
the 2 x 2 case. But, there are four more terms, and we need to understand
them. The animation below shows how the determinant is built from
the elements within the matrix.
Note the following about this computation:

In each term in the result,
one coefficient is chosen from each row.

In the aei
term,

a is in the first row,

e is in the second row
and

i is in the third row.

In each term in the result,
one coefficient is chosen from each column.

In the aei
term,

a is in the first column,

e is in the second column
and

i is in the third column.

Those conclusions about
rows and columns are true for every term in the result.

The result contains every
possible way to choose one element from the first row, one from the second
row, etc., and one from the first column, one from the second column, etc.,
without ever choosing two terms from the same row or column for any single
term.

The result can be interpreted
as a sum of all the possible ways to choose terms from the main descending
diagonal, and two subdiagonals. For example, the bfg term
has b and f along a short "diagonal", and that term picks up the g term.
The cdh term also has two coefficients along a short diagonal
 d and h  and one coefficient  c  to fill out that term.

There is a system to the
way signs are assigned. Actually, the algorithm is that the sign
depends upon whether the permutation of the coefficient indices is odd
or even.

We will leave that to
a math textbook.
Here
is the animation again. Check the claims above again.
Now,
what about the case of a 4 x 4 determinant? Consider the following.

There are four
ways to choose an element from the first column.

After an element is chosen
from the first column, there are three rows left from which an element
could be chosen. Thus, there are three
ways to choose an element from the second column.

After an element is chosen
from the second column, there are two rows left from which an element could
be chosen. Thus, there are two
ways to choose an element from the third column.

That leaves one
way to choose an element from the fourth column.
Putting
that all together, we can compute the number of terms in the expression
for the determinant.
Number of Terms
in the Determinant = 4 x 3 x 2 x 1 = 24 terms!

Conclusion:

There is no simple
algorithm to visualize the determinant of a 4 x 4 matrix!
Problems
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your comments on these lessons.