2.4.7 Using Cramer's Rule

Recall that in the introduction to Section 2.4 we stated that not all of the four methods we are studying are computationally efficient. With our fourth and final computational method, Cramer's Rule, we will illustrate the prime example of a very poor computational tool from which we can learn much.

Cramer's Rule is stated in terms of the determinant of the coefficient matrix and the determinants of some matrices derived from the coefficient matrix. A review of determinants can be found in Section 7.6.

To get an intuitive feel for why Cramer's Rule works and the role of determinants in Cramer's Rule, we will start with its derivation for a 2 x 2 system of linear equations. Consider the following system of equations:

(2.12)

(2.13)

(2.14)

Now let's write the system (2.12) in matrix form:

Now we may write the equations for x₁ and x₂ in terms of the determinants of A, B₁, and B₂.

for i = 1,2. This is Cramer's Rule for a 2 x 2 system.

Cramer's Rule for General Linear Systems.

An Example.

We can apply Cramer's Rule to the example we first introduced at the beginning of this chapter. Recall that the system of equations that Professor Tapia derived to determine the number of time zones separating New York and Geneva was

(2.15)

Cramer's rule is easy to remember and perform on a 2 x 2 system like this one (since it can be written in closed form, i.e. as a formula), but it is not so easy to perform as the size of the system increases. For an n x n system, we must compute n + 1 determinants (|A|, |B₁|, |B₂|, ... , |B_n|). If we increase the size of the system to (n + 1) x (n + 1), then we will have to compute n + 2 determinants instead of n + 2, and all the determinants we compute will be of larger matrices ( (n + 1 x (n + 1) instead of n x n). Hence the amount of effort required to solve a system of linear equations using Cramer's rule increases significantly as the size of the system increases.