Cramer’s Rule

Definition

Cramer's Rule is a mathematical theorem used to solve systems of linear equations with as many equations as unknowns, utilizing determinants. It provides an explicit formula for the solution of the system.

5 Must Know Facts For Your Next Test

1. Cramer's Rule can only be applied to square matrices where the number of equations equals the number of unknowns.
2. The rule involves calculating determinants of the coefficient matrix and matrices formed by replacing one column with the constants from the equations.
3. If the determinant of the coefficient matrix is zero, Cramer's Rule cannot be applied because it indicates that there is either no unique solution or infinitely many solutions.
4. For a system of $n$ linear equations, $n+1$ determinants need to be calculated: one for the coefficient matrix and one for each variable's matrix.
5. Cramer's Rule is computationally expensive for large systems due to determinant calculation complexity.

Review Questions

• What condition must be met for Cramer's Rule to be applicable?
• How do you form the matrices needed to apply Cramer's Rule?
• What does it imply 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 that matrix, such as whether it is invertible.

Coefficient Matrix:

A matrix consisting solely of the coefficients of variables in a system of linear equations.

Linear Equation:

An equation that makes a straight line when graphed, typically written in the form $ax + by + cz = d$.