Matrix Calculator: Complex Matrix & Linear System Solver

Use this free online matrix calculator to perform basic matrix operations—addition, subtraction, multiplication, and division (by inversion)—on complex matrices. By applying elementary row operations, the calculator can also perform advanced matrix operations, including finding the determinant, inverse, and adjugate for n×n (square) matrices. Also, for any m×n matrix, you can find the rank or transform the matrix into lower/upper triangular forms and reduced row echelon form (RREF).

Additionally, you can solve systems of linear equations with a complex augmented matrix via Gauss-Jordan elimination. You can also use our unique matrix expression calculator to evaluate sophisticated formulas involving up to eight matrices simultaneously.

Quick Start

One Interface for All Matrices

This innovative matrix calculator and linear system solver provides a single, dynamic interface to conveniently manage multiple matrices, including augmented matrices for linear systems of equations.

Set Up Your Matrix

  • Select: Pick a Matrix (A, B, C, D, E, F, G, H) from the dropdown menu.
  • Resize: Use the ＋ buttons or type dimensions directly into the input boxes.
  • Input: Enter numbers or constant expressions (e.g., 5/3 or (2+3i)sin(pi/2)).
  • Randomize: Use the Random button to instantly populate a matrix with test data.

Convenient Matrix Operations

Use the buttons at the top of the calculator to perform the following operations:

  • det: Matrix Determinant
  • inv: Matrix Inverse (or Inverse Matrix)
  • adj: Matrix Adjugate (or Adjugate Matrix)
  • rank: Matrix Rank
  • rref: Reduced Row Echelon Form
  • ...and more

Use the Quick Calculations dropdown for common tasks like A*B or A+B.

Custom Expressions

Type formulas directly into the Expression Box:

  • Multiplication: ABC
  • Combined: 2A + inv(BC) - D^3

Solve Linear Systems (e.g., Ax = b)

Check the Linear System box. This turns the matrix interface into an Augmented Matrix by adding a highlighted column for the column vector (b).

  • Fill in the coefficients (A) and the column vector (b).
  • Press Solve to view the solution and the RREF.

Note: The matrix calculator automatically saves dimensions, entries, and augmented settings for all matrices between visits. To wipe all data and start fresh, use the Reset Calculator button.

Matrix Calculator Complete Instructions

About the Matrix Calculator

This advanced matrix calculator is designed to efficiently perform matrix operations on up to 10×10 matrices. It fully supports complex matrices—those containing real, imaginary, or complex entries.

The calculator automatically verifies dimension compatibility for all matrix calculations and provides detailed error messages when requirements are not met.

Matrix Operations

Beyond elementary matrix operations, the calculator performs advanced operations on square matrices (e.g., finding the determinant or inverse). For matrices of any dimension, you can perform other advanced operations such as finding rank and RREF.

Matrix Expression Handling

What sets our calculator apart is its ability to evaluate intricate matrix expressions involving up to eight compatible matrices, such as: A+BCD-inv(E)

Solve Systems of Linear Equations

Our linear system solver uses Gaussian elimination to solve systems of linear equations by performing row operations on augmented matrices. As a complex augmented matrix calculator, it fully supports systems involving a complex coefficient matrix and complex vector column.

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Results

Elementary & Advanced Matrix Algebra

Suitable for all levels, this easy-to-use, powerful complex matrix calculator simplifies and evaluates expressions involving complex matrices and performs matrix operations ranging from simple arithmetic to advanced linear algebra.

Basic Matrix Operations

Perform basic matrix algebra with our online calculator:

Advanced Matrix Operations

Determinant, Inverse, and Adjugate

Calculate the following properties for square matrices:

Rank, Lower/Upper Triangular Forms, RREF, and Transpose

Calculate the following for any matrix: