A Short Discourse On Determinants

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, N₁, N₂ 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, N₁ and N₂, are computed in the same way, but not with the same coefficients. N₁ is similar to D but for N₂ the first column is replaced by the values on the right hand side of the set of equations. This means:

So, we find: N₁ = 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:

x = (ed - fb)/(ad - cd)

We can also compute the solution for y. However, in the set of equations y appears in the second column, so to compute N₂ 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 sub-diagonals. 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

Send your comments on these lessons.