## Cramer's Rule

Recall the following:

• The linear system has a unique solution for every if and only if exists.
• has an inverse if and only if
Thus, has a unique solution FOR EVERY if and only if

The following theorem gives a direct method of finding the solution of the linear system when

THEOREM 2.6.18 (Cramer's Rule)   Let be a linear system with equations in unknowns. If then the unique solution to this system is

where is the matrix obtained from by replacing the th column of by the column vector

Proof. Since Thus, the linear system has the solution Hence, the th coordinate of is given by

height6pt width 6pt depth 0pt

The theorem implies that

and in general

for

EXAMPLE 2.6.19   Suppose that and Use Cramer's rule to find a vector such that
Solution: Check that Therefore
and That is,

A K Lal 2007-09-12