Advanced Engineering Mathematics, 7th Edition

8.7 Cramer’s Rule

INTRODUCTION

We saw at the end of the preceding section that a system of n linear equations in n variables AX = B has precisely one solution when det A ≠ 0. This solution, as we shall now see, can be expressed in terms of determinants. For example, the system of two equations in two variables

a₁₁x₁ + a₁₂x₂ = b₁

a₂₁x₁ + a₂₂x₂ = b₂ (1)

possesses the solution

(2)

provided that a₁₁a₂₂ − a₁₂a₂₁ ≠ 0. The numerators and denominators in (2) are recognized as determinants. That is, the system (1) has the unique solution

(3)

provided ...