# 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

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.

[Geometry] [Algebra] [Trigonometry ]
[Calculus] [Differential Equations] [Matrix Algebra]

Do you need more help? Please post your question on our S.O.S. Mathematics CyberBoard.

Author: M.A. Khamsi