Subject: order 4+ determinants
Name: Joe Kron
Who is asking: Other
Level: Secondary

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!


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