On-Line Geometric Modeling Notes
CRAMER'S RULE

Overview

Cramer's Rule is a determinant-based procedure utilized to solve systems of equations. In these notes we first discuss Cramer's Rule for systems of three linear equations with three unknowns and then state the Cramer's rule for general systems of equations.

Cramer's Rule - Three Equations, Three Unknowns

Given a system of three linear equations, with three unknowns,

Cramer's Rule depends on the calculation of four determinants. If we define to be the determinant defined by

and the determinants , and , to be

respectively, then Cramer's rule states that

Note that the determinants are easily generated. The columns of are just the coefficients of , and respectively in the linear equations and is produced by replacing the th row by the row

Example

Suppose we are given a point in space and a frame . By the definition of a frame, the point can be written

and suppose we wish to find the coefficients so that this is true. Now, is a vector, and so we can write down the equations

where the is the cartesian coordinate representation of , is the representation of , is the representation of and is the representation of . By Cramer's Rule, we can solve this by defining the following four determinants

and then

The Homogeneous Case

Given a system of three linear equations, with three unknowns,

then the system is called homogeneous if . In this case, if , then Cramer's rule above gives the solution - the trivial solution. However, if , then the rank of the coefficient matrix is less than 3, and the system will have non-trivial solutions.

The Example without Determinants

If we look closely at the determinants in the above example, we can see that they can actually be expressed in terms of the vectors , , , and . In particular,

Thus, in this case, we can eliminate the determinants and utilize dot and cross products.

If Everything is Nice

It is worth looking at the vector-based Cramer's rule one more time for the case when the frame is orthonormal. If the vectors are all mutually perpendicular and of unit length then the above equations simplify significantly . In particular, if we assume that

which is the case in a right-handed orthonormal system, we have

Summarizing this case, it is clear that we only need dot products to calculate the determinants.

No cross products are required.

The General Cramer's Rule

Given a system of linear equations

If the determinant of the coefficient matrix is not zero then the system has precisely one solution. This solution is given by the formulas

where is the determinant obtained from by replacing the th row of by the row with the entries

Also, if the system is homogeneous and , then it has only the trivial solution . If , the homogeneous system has nontrivial solutions.

Ken Joy
2000-11-28