5.18. Cramer’s Rule

Theorem 5.18:  Cramer’s Rule.

Consider a system of n linear equations

        j=1 n a ij x j = b i                               ( i=1,,n  )

with n unknowns x 1 ,, x n .  If the coefficient matrix A=( a ij )  is nonsingular, the system has a unique solution given by

        x j = 1 detA k=1 n b k cof a kj              for  j=1,,n                (5.31)

Proof

The system can be written as a matrix equation

        AX=B

where X= ( x 1 ,, x n ) t  and B= ( b 1 ,, b n ) t  are column matrices.  Since A is nonsingular, there is a unique solution

        X= A 1 B   = 1 detA ( cofA ) t B                    (5.32)

Þ            x j = 1 detA k=1 n ( cofA ) kj b k = 1 detA k=1 n b k cof a kj                 QED.

Comment

Note that k=1 n b k cof a kj  is the determinant of a matrix Cj obtained from A by replacing the jth column with B.  [See Theorem 5.20 for the formula for expansion by column.]  Thus, (5.31) can be put into the perhaps better known form

        x j = det C j detA