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 x_i = 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:

x_i = 1/det(A)*∑(j=1,n) ent_i,j(adj A))b_j

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

= 1/det(A)*∑(j=1,n) b_j*(cof_j,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)b_j*(cof_j,i(A)) = 1/det(A)*det(A_{(Ci ↔ B)}).

(7) Putting it all together gives us:

x_i = 1/det(A)*det(A_{(Ci ↔ B)}).

QED

References

• Charles G. Cullen, Matrices and Linear Transformations, Dover Publications, Inc., 1972.
• "Cramer's Rule", PlanetMath.org