Monday, September 17, 2007

Cramer's Rule

In today's blog, I go over the classic result known as Cramer's Rule.

Theorem: Cramer's Rule

Let an n x n matrix A represent a system of linear equations such that AX = B.

Then it follows that if Det(A) ≠ 0, then X has only one unique solution and xi = det(A(Ci ↔ B))/det(A).

Proof:

(1) Assume that Det(A) ≠ 0

(2) Then A is invertible [see Theorem 4, here]

(3) Now, A-1B is a unique solution to AX = B [See Lemma 2, here]

(4) From Corollary 4.1, here, we have:

X = A-1B = 1/det(A)(adj A)B

(5) Therefore, we have:

xi = 1/det(A)*∑(j=1,n) enti,j(adj A))bj

[See Definition 1 here for definition of matrix multiplication if needed]

= 1/det(A)*∑(j=1,n) bj*(cofj,i(A)) [See Definition 1 here for definition of adj A]

(6) Using Corollary 4.1 here, this gives us:

1/det(A)*∑(j=1,n)bj*(cofj,i(A)) = 1/det(A)*det(A(Ci ↔ B)).

(7) Putting it all together gives us:

xi = 1/det(A)*det(A(Ci ↔ B)).

QED

References

3 comments :

Infinity said...

Thank you very much sir!

Your posts and your blog "Math Refresher" is a real help for me in my course. You are great at explaining concepts! :P

www.algebrasolver.totalh.com said...

yeah "Math Refresher" is really good at explainning concepts, I was looking for something simple like this :P

www.algebrasolver.totalh.com said...

in my last course of linar algebra, cramer's rule got in a real mess for me...