Page 100 - A Course in Linear Algebra with Applications

84 Chapter Three: Determinants

Cramer's Rule
For a second illustration of the uses of determinants, we
return to the study of linear systems. Consider a linear system
of n equations in n unknowns x\, X2,..., x n

AX = B,

where the coefficient matrix A has non-zero determinant. The
1
system has a unique solution, namely X = A~ B. There is a
simple expression for this solution in terms of determinants.
Using 3.3.7 we obtain

l
X = A~ B = l/det(A) (adj(A) B).

From the matrix product adj(^4)JB we can read off the ith.
unknown as

n n
Xi = (5>dj(A))^)/det(A) = C^bjAjJ/detiA).

Now the second sum is a determinant; in fact it is det(Mj)
where Mi is the matrix obtained from A when column i is
replaced by B. Hence the solution of the linear system can be
expressed in the form Xi = det(Mj)/det(^4), i = 1,2, ...,n.
Thus we have obtained the following result.
Theorem 3.3.8 (Cramer's Rule)
If AX — B is a linear system of n equations in n unknowns
and det (A) is not zero, then the unique solution of the linear
system can be written in the form

Xi = det(Mi)/det(j4), i = l, ... , n,

where Mi is the matrix obtained from A when column i is
replaced by B.