Matrix Calculator: Complex Matrix & Linear System Solver

Use this free online complex matrix calculator to perform matrix algebra and solve systems of linear equations by performing Gaussian elimination on augmented matrices with real and imaginary number entries.

Quick start

To fill or edit a matrix (A, B, C, ..., H), simply select it from the drop-down list.
You can add rows or columns to a selected matrix by pressing the ＋ (insert row or insert column) buttons provided.
You can also change the dimensions of a selected matrix by entering the number of its rows or columns in the text boxes provided or by conveniently pressing the numbered buttons on the left of a row or above a column, respectively.
The matrix calculator remembers the dimensions and entries of all matrices you enter, as well as whether a matrix is augmented for solving systems of linear equations.

About the Complex Matrix Calculator

This powerful matrix calculator is designed for matrix operations involving complex matrices—those with real, imaginary, or generally complex number elements. What sets our calculator apart is its ability to evaluate intricate matrix expressions, handling up to eight matrices, for example: A+ABC-inv(D) as long as they have compatible m×n (m by n) dimensions. Furthermore, all matrices can be entered via a single interface.

One Interface for All Matrices

This innovative matrix & linear system solver provides a single, dynamic interface for entering multiple matrices, including augmented matrices that represent systems of linear equations.

Solve Systems of Linear Equations

This augmented matrix calculator seamlessly solves linear systems of equations using Gaussian elimination. It performs row operations on augmented matrices, reducing them to a form where solutions are easily identified. As a powerful linear system solver, it can handle augmented matrices that are also complex matrices.

Show work

Linear System

Results

Instructions for the matrix calculator

Basic and Advanced Matrix Operations

Beyond basic operations of matrix addition, matrix subtraction, and matrix multiplication, our matrix calculator can compute the following for n×n square matrices (e.g., 2×2, 3×3, 4×4):

Matrix determinant
Matrix inverse
Matrix adjugate

Additionally, you can compute the following for any matrix, real or complex:

Matrix rank
Matrix reduced row echelon form (RREF)
Upper and lower triangular forms
Matrix Transpose

Matrix Calculator as a Multi-Variable Function Evaluator

You can also utilize the matrix calculator as a multi-variable function evaluator.

Type in a function expression containing up to 8 variables in the matrix expression box, using A,B,C,... as variables (instead of x, y, z, ...).
Set all matrices involved as 1×1 matrices, filling them with numbers or constant expressions.
Press the Calculate button. The value of the function is given as a scalar.