Page 137 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

THEOREM 2.1.4 , then the system has a unique solution.

Cramer's Rule
If is a system of linear equations in unknowns such that
This solution is

where is the matrix obtained by replacing the entries in the th column of A by the entries in
the matrix

Proof If , then A is invertible, and by Theorem 1.6.2, is the unique solution of . Therefore, by

Theorem 2.1.2 we have

Multiplying the matrices out gives (11)

The entry in the th row of is therefore
Now let

Since differs from A only in the th column, it follows that the cofactors of entries , , , in are the same as the

cofactors of the corresponding entries in the th column of . The cofactor expansion of along the th column is

therefore

Substituting this result in 11 gives