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:

        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:

        Here is the animation again.  Check the claims above again.
        Now, what about the case of a 4 x 4 determinant?  Consider the following.         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!


Problems
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