The cyclic determinants are most properly in these forms:
Here comes the most important point : Det(X) is NOT symmetrical, it's asymmetrical because
But it's cyclic since putting f(a,b,c) to f(b,c,a) involve two times of interchanging rows, we will skip this part.
At the same time, Y is symmetrical because
And now we will discuss an alternative technique in factorizing determinants.
By factor theorem, if f(a)=0, then (x-a) is a factor of f. Similar we state determinants as a polynomial function, for 2 variables a,b in the determinants which a=b (or other condition), then (a-b) is a factor of it.
Example 1: Factorize
This is the traditional approach:
Some students think that it's difficult to reach the idea to add up the rows, so you can factorize some easier component first:
Since when x=y, the three rows (and at least two of it) are identical. Therefore (x-y) must be a factor of the determinant.
We used different approach, but the answer's still unique.
In this example I've demonstrated in a symmetrical-like (even some of its factor is not symmetrical) determinants how factorization can be easier.
Example 2: Factorize
This is a classic example. I have proven that this is asymmetrical, i.e., X=0 if a=b or b=c or c=a. Therefore (a-b)(b-c)(c-a) is a factor of x. Equating the degree,
For determinants in this form it can be factorized into one product nicely. By measuring its degree and asymmetrical properties, factorizing can be much easier.
Exercises:
1) Solve the following problems about determinants:
a) Show without expanding that
(Hint: consider the transpose of determinants)
b) Show that
c) Factorize
d) Show without expanding that
e) Show without expanding that
f) Factorize
Reference:
1) My project(2010): Basic Algebraic Skills, p.46-49 "Multivariate function". You're also welcomed to finish the exercise there.
2) Margaret M. Gow(1960), A Course in Pure Mathematics p.33, problems may or may not modified from here.
These problems about determinants are actually quite hard, so if you can do them without difficulties, probably you have mastered this topic!
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