Subject: order 4+ determinants

Name: Joe Kron

Who is asking: Other

Level: Secondary
Question:

Why is it never shown how to calculate the value of 4x4 (or larger size) deteminants by the diagonal multiply methods that are generally shown for 2x2 and 3x3 determinants? The method I'm talking about is called Cramer's Rule??? Is this method not extensible to order 4+ and if not why not? Anyway the method always shown for order 4+ is called "reduction by minors" which is not the answer to this question.

Hi Joe,

The simple answer is: there IS no 'diagonal' method for 4x4 plus
determinants.

Sure Cramer's Rule works for something - but that is a method for SOLVING,
say a 3x3 system of equations A X = B, using a set of four
3x3 determinants. It does NOT find a 4x4 determinant.

Reduction by minors, or Laplace decomposition, is the standard way,
in various forms, of calculating larger determinants.
If you look at that method, you will confirm that, expanded out,
you need 24 terms (multiplications of four entries, one from each column
and each row) to be added. There is no pattern of diagonals, even with
added columns or rows outside (as you did for 3x3) that gives you
all these terms.

Now the reality is that people do NOT take large determinants,
by hand. They are a really important IDEA. We reason about what
can happen with systems of equations using determinants.
I have even seen, occassionally, a 36x36 'determinant' worked out
by hand. In that case, because there were lots of zero entries,
it was possible to keep it simple enough to get an answer.

The process is what people in computer science call 'exponential'.
The number of steps explodes as you increase the size, growing
like an exponential function (actually n! for an nxn matrix).
If you really have to work with such a calculation - that is what
computers are for!

Again, I repeat the fact that the CONCEPT is important.
I use it all the time to describe patterns and calcuations I do.
This is for work on structural rigidity of frameworks,
reconstructing 3-D objects from plane pictures,
programming for CAD designs, etc. I note that computer
graphics for 3-D reasonably involves 4x4 matrices.

The main reason someone
may actually test you on 5x5 or higher matrices will be to
see if you understand the principles, and can reason about information
that comes from the determinant. It is something to understant.
It is not something to try to invent rapid, repeated calculations
about!