. . (ii) Cramer’s Rule This rule can be applied only when the coefficient matrix is a square matrix and non-singular. It is explained by considering the following system of equations: a x a x a x a x a x a x a x a x a x = b , where the coefficient matrix is non-singular.
Then ¹ . Let us put D = . Then, we have x D = x a x a x a x a x a x a x a x a x a x a x a x a x = ∆ - - Since D ¹ , we get x = ∆ ∆ Similarly, we get x = where ∆ ∆ = ∆ ∆ ∆= ∆= b a b a b a Thus, we have the Cramer’s rule x = ∆ ∆ = ∆ ∆ = ∆ ∆ where D = b a b a ∆= b a b a b a b a ∆= ∆= Note Replacing the first column elements a of D with b b b respectively, we get D . Replacing the second column elements a of D with b b b respectively, we get D .
Replacing the third column elements a of D with b b b respectively, we get D . If ∆= , Cramer’s rule