Guide to Using Our Matrix Calculator

Matrix Selection & Editing

To fill a matrix with entries, select a named matrix (A, B, C, ..., H) from the drop-down list. Initially, all matrices are 2×2 and filled with zeros, except for matrix A.
If necessary, press the Clear button to delete the entries of a selected matrix.

Setting Matrix Dimensions

You can set the dimensions of a selected matrix in the following ways:

Increasing Dimensions: Add rows or columns by pressing the ＋ (insert row/insert column) buttons.
Direct Selection: Set the number of rows or columns by clicking the numbered buttons to the left of a row or above a column.
Direct Entry: Enter the number of rows or columns in the provided text boxes.

Entering Matrix Data

You can fill matrices with real numbers, imaginary numbers, complex numbers, or expressions like 1/2 + 3sin(5π/4)i.

The calculator remembers the dimensions and entries of all matrices, including whether a matrix is augmented.
All data is retained indefinitely. To clear all data, press the Reset Calculator button.
You can also fill a selected matrix with random integers between -10 and 10 using the provided button.

Matrix Algebra

To calculate various matrix operations such as determinant, inverse, reduced row echelon form (RREF), adjugate, rank, lower/upper triangular forms, and transpose on a selected matrix, press the corresponding buttons at the top of the matrix calculator.

Quick Calculations & Custom Expressions

Under the Quick Calculations drop-down list, you can compute frequently used matrix expressions involving two or more matrices, such as A+B, (A+B)(C+D), and more.

If a desired matrix expression is not listed in the Quick Calculations menu, enter it in the expression box and press Calculate.

You can use the following in your expressions:

inv() – Inverse (used for A^-1)
det() – Determinant
adj() – Adjugate
transp() – Transpose
rref() – Reduced Row Echelon Form
ut() – Upper Triangular Form
lt() – Lower Triangular Form

If the expression is valid and contains no incompatible matrix operations, the result will be displayed. Otherwise, an error message will appear.

As an example, the calculator is capable to evaluate intricate matrix expressions such as (2+sin(π/3))A+inv(A+iB/det(A))(B/2-BC^4)/D^(2^3+1)

Remark: Although division is not a standard mathematical operation on matrices, our matrix calculator allows the use of the division operator as in A/B in matrix expressions, interpreting it as A*inv(B) for compatible matrices where B is invertible. Additionally, note that 1/A can be used for inv(A).

1×1 Matrices & Scalar Treatment

All 1×1 matrices are treated as scalars and can be multiplied by any matrix, regardless of dimension.
For example, if A = [1/2], then sin(A) is treated as sin(1/2).
Conversely, scalars can be treated as 1×1 matrices; that is, inv(2) is the same as inv([2]), which results in 0.5.

Solving Linear Systems

The system of linear equations solver provides two methods for solving such systems.

Method 1: Solving a Single Linear System

The first method is a straightforward approach for solving a single linear system.

Select the Linear System checkbox.
Set the dimensions for the coefficient matrix and fill in the augmented matrix with real or complex numbers (or expressions).
The last column (column vector) represents the right-hand side (RHS) of the system, highlighted with a different background.
Press Solve.
- If the system is consistent and has a unique solution, the solution vector will be displayed along with the Reduced Row Echelon Form (RREF) of the augmented matrix.
- If multiple solutions exist, the general solution will be provided.
- If the system is inconsistent, a message indicating this will appear, along with the RREF.

Method 2: Solving Multiple Systems with the Same Coefficient Matrix

This method allows you to solve multiple linear systems simultaneously, all sharing the same m×m square matrix as their coefficient matrix.

Do not select the Linear System checkbox.
Set the number of columns to be greater than the number of rows (m).
Fill each additional column (after the mth column) with the right-hand side (RHS) of the corresponding system.
Press RREF.
- If the m×m matrix on the left appears in RREF, each system has a unique solution.
- The corresponding solution vectors will be displayed on the right-hand side.