28.1. THE DETERMINANT 535

28.1.9 Cramer’s RuleIn case you are solving a system of equations, Ax= y for x, it follows that if A−1 exists,

x=(A−1A

)x= A−1 (Ax) = A−1y

thus solving the system. Now in the case that A−1 exists, there is a formula for A−1 givenabove. Using this formula,

xi =n

∑j=1

a−1i j y j =

n

∑j=1

1det(A)

cof(A) ji y j.

By the formula for the expansion of a determinant along a column,

xi =1

det(A)det

∗ · · · y1 · · · ∗...

......

∗ · · · yn · · · ∗

 ,

where here the ith column of A is replaced with the column vector (y1 · · · ,yn)T , and the

determinant of this modified matrix is taken and divided by det(A). This formula is knownas Cramer’s rule.