Solving systems with matrices

Math

Definition

Using matrix operations to solve systems of linear equations. The system is written in matrix form Ax = b, and solutions can be found using inverse matrices (x = A⁻¹b), row reduction (Gaussian elimination), or Cramer's rule with determinants.

How It Works

  1. Write the system of equations as an augmented matrix [A|b].
  2. Use row operations to reduce to row echelon or reduced row echelon form.
  3. Back-substitute to find each variable.
  4. Alternatively, compute A⁻¹ and multiply by b to get x = A⁻¹b.

Examples

  • Solving 2x + y = 5, x − y = 1 using an augmented matrix and row reduction
  • Using a graphing calculator's matrix function to solve a 3×3 system

Related Terms

Matrix operations Determinants Inverse matrices Systems of three variables

Study This Concept

Practice solving systems with matrices with free review games in these units:

Pre-Calculus Unit 7: Systems and Matrices