Cramer’s Rule

Definition

Cramer's Rule is a method for solving systems of linear equations using determinants. It provides an explicit solution for each variable by dividing the determinant of a modified matrix by the determinant of the coefficient matrix.

5 Must Know Facts For Your Next Test

1. Cramer's Rule can only be applied when the system has the same number of equations as unknowns and the coefficient matrix is invertible.
2. The rule involves calculating determinants, which are scalar values derived from square matrices.
3. If the determinant of the coefficient matrix is zero, Cramer's Rule cannot be used because it indicates that the system does not have a unique solution.
4. To find each variable using Cramer's Rule, you replace one column of the coefficient matrix with the constants from the right-hand side of the equations and calculate its determinant.
5. The formula for solving for a variable $x_i$ in a system is given by $x_i = \frac{D_i}{D}$ where $D$ is the determinant of the coefficient matrix and $D_i$ is the determinant of the modified matrix.

Review Questions

• What condition must be met for Cramer's Rule to be applicable?
• How do you modify a matrix to use Cramer's Rule to solve for a specific variable?
• Why can't Cramer's Rule be used if the determinant of the coefficient matrix is zero?

Related terms

Determinant: A scalar value that can be computed from a square matrix and encodes certain properties of the matrix.

Coefficient Matrix: A matrix consisting solely of coefficients from a system of linear equations.

Inverse Matrix: A matrix that, when multiplied with its original matrix, yields an identity matrix.