Systems of Linear Equations: Cramer's Rule

Definition: Every square n × n matrix A has a real number associated with it called the
determinant of A, denoted by |A|.

Definition: Determinant of a 2 × 2matrix:

Example (1) : Evaluate

solution :

Definition: The minor M_ij of a square n × n matrix A is the determinant of the matrix
found by deleting the i^th row and j^th column of A.

Example (2): Evaluate M₂₃ where

solution : Delete the second row and third column then evaluate the determinant of the
resulting 2 × 2 matrix.

Definition: Determinant of a 3× 3 matrix:

Example (3): Evaluate the determinant of the matrix from example (2)

In example (3) we calculated the determinant by expanding across the first row. We can
calculate the determinant by expanding across any row or any column by using the
following sign array.

Example (4): Calculate the following determinant by expanding across the second
column.

Note: When calculating the determinant it is always best to expand about the row or
column with the greatest number of 0’s.

Theorem: Cramer’s Rule for Two Equations and Two Variables
The solution to the system of equations

is given by

provided D ≠ 0

Examples: Solve using Cramer’s Rule:

solution :

the solution to the system is given by

A similar procedure is used for solving a system of three equations and three variables.

solution :

The solution is