Application of Determinant to Systems: Cramer's Rule
We have seen that determinant may be useful in finding the inverse of a
nonsingular matrix. We can use these findings in solving linear systems for
which the matrix coefficient is nonsingular (or invertible).
Consider the linear system (in matrix form)
A X = B
where A is the matrix coefficient, B the nonhomogeneous term, and
X the unknown column-matrix. We have:
Theorem. The linear system AX = B has a unique solution
if and only if A is invertible. In this case, the solution is given by
the so-called Cramer's formulas:
where xi are the unknowns of the system or the entries
of X, and the matrix Ai is obtained from A
by replacing the ith column by the column B. In
other words, we have
where the bi are the entries of B.
In particular, if the linear system AX = B is homogeneous,
meaning
 ,
then if A is invertible, the only solution is the trivial one, that is
 .
So if we are looking for a nonzero solution to the system, the matrix
coefficient A must be singular or noninvertible. We also know that this
will happen if and only if
 .
This is an important result.
Example. Solve the linear system
Answer. First note that
which implies that the matrix coefficient is invertible. So we may use the
Cramer's formulas. We have
We leave the details to the reader to find
Note that it is easy to see that z=0. Indeed, the determinant which gives
z has two identical rows (the first and the last). We do encourage you to
check that the values found for x, y, and z are indeed the
solution to the given system.
Remark. Remember that Cramer's formulas are only valid for linear
systems with an invertible matrix coefficient.
|
