Cramer's rule

Cramer's rule is a method for solving square linear systems using determinants. It works only when the system has a unique solution.

To solve a square system by Cramer's rule

Given a system Ax = b, where A is square and invertible:

Find det(A) (which must be non-zero).

For variable number k, replace the kth column of det(A) by b and evaluate this new determinant.

The kth variable is this last determinant divided by det(A).

Here's a picture of Cramer's rule for 3 x 3 square linear systems.

Remember, Cramer's rule doesn't work if the determinant of the coefficients is 0, so calculate that one first.

Cramer's rule is very inefficient for systems much larger than 2 x 2. Its uses are mainly theoretical, or in situations where the determinants have some special form, or if you need to find only one of the variables.

For example, if

2x + 3y = 4
x – 5y = 2

and you only need to find x, then

.