Matrix Calculator

Name: Matrix Calculator
Author: Jitendra Kumar

Calculate matrix multiplication, addition, subtraction, scalar multiplication, transpose, determinant, inverse, and RREF with dimension checks.

Last Updated: June 13, 2026

Matrix operation

Enter rows on separate lines. Separate entries with spaces, commas, semicolons, or tabs.

Matrix A

Matrix B

A x B

3 x 3

A Dimensions

3 x 3

B Dimensions

3 x 3

Trace

Result matrix

10	3	14
28	9	32
47	15	52

Detail	Value
Formula	c_ij = sum(a_ik x b_kj)
Rows in result	3
Columns in result	3
Minimum entry	3
Maximum entry	52

Step	Explanation
1	A is 3 x 3 and B is 3 x 3, so the product is 3 x 3.

Dimensions

Addition requires matching dimensions. Multiplication requires columns of A to match rows of B.

Determinants

Determinants and inverses only apply to square matrices.

RREF

RREF uses row operations and is useful for solving systems and checking rank.

Numerical Linear Algebra Notice

This calculator uses decimal arithmetic and rounding for display. Very ill-conditioned or nearly singular matrices may need exact symbolic tools or specialized numerical software for final coursework, engineering, or research use.

Checked by Jitendra Kumar

Matrix Calculator is checked for formula labels, source links, and result limits.

Jitendra Kumar, Founder & Editorial Standards Lead. Updated June 13, 2026. Scope: math calculators.

Sources & methodology · Review standards

How to Use the Matrix Calculator

Choose the matrix operation, then paste Matrix A and Matrix B when needed. Put each row on its own line and separate entries with spaces, commas, semicolons, or tabs.

Use determinant and inverse modes for square matrices. Use RREF mode for row reduction or augmented systems where constants appear in the final column.

Step 1: Choose an operation
Select addition, subtraction, multiplication, scalar multiplication, transpose, determinant, inverse, or RREF.
Step 2: Enter matrix rows
Type each row on a new line and separate entries consistently.
Step 3: Check dimensions
Use the result cards and error messages to confirm the operation is valid.
Step 4: Review the output
Read the result matrix, scalar output, formula, and row-operation notes.

How This Matrix Calculator Works

Element-wise operations combine matching matrix entries. Matrix multiplication uses row-by-column dot products, so the inner dimensions must match.

Determinant, inverse, and RREF calculations use Gaussian-elimination style row operations. Inverse mode augments Matrix A with the identity matrix and row-reduces to isolate the inverse.

Results are rounded for readability, and very small values near zero are displayed as zero to reduce numerical noise.

Matrix Operation Guide

Matrix Operation Rules

Operation	Requirement	Meaning
Addition / subtraction	same dimensions	Combine matching entries element by element.
Multiplication	columns of A = rows of B	Each result entry is a row-by-column dot product.
Scalar multiply	any matrix	Multiply every entry by the same scalar.
Transpose	any matrix	Rows become columns and columns become rows.
Determinant	square matrix only	Returns a scalar used for invertibility and scaling checks.
Inverse	square, nonsingular matrix	A times A inverse equals the identity matrix.
RREF	any rectangular matrix	Uses row operations to simplify a matrix or augmented system.

Input Format

Input part	Rule	Example
Rows	Put each matrix row on its own line.	Example: first row on line 1, second row on line 2.
Columns	Separate entries with spaces, commas, semicolons, or tabs.	Example: 1 2 3 or 1, 2, 3.
Decimals	Decimal entries are accepted.	Results are rounded for display.
Augmented systems	Use RREF mode with constants as the final column.	Useful for linear systems.

Matrix Formula Examples

Task	Formula or setup	Example result
2x2 determinant	det([[a,b],[c,d]]) = ad - bc	[[3,2],[5,4]] gives 3 x 4 - 2 x 5 = 2.
Matrix product entry	c11 = row 1 of A dot column 1 of B	[1,2] dot [3,4] gives 11.
Scalar multiplication	kA multiplies every entry by k	3 x [[1,-2],[0,4]] = [[3,-6],[0,12]].
Inverse check	A x A inverse = identity matrix	If the product is not close to identity, inspect rounding or singularity.

Most matrix mistakes are dimension mistakes. Check rows and columns before doing multiplication, and remember that A x B usually differs from B x A.

For equation-level algebra, use the math equation solver. For numeric checks around entries or determinants, use the scientific calculator.

Keep the research moving with Scientific Calculator, Math Equation Solver, Statistics Calculator, and Probability Calculator.

Frequently Asked Questions

It supports addition, subtraction, multiplication, scalar multiplication, transpose, determinant, inverse, and reduced row echelon form.

Matrix A can multiply matrix B when the number of columns in A equals the number of rows in B.

A matrix must be square and nonsingular to have an inverse. If the determinant is zero or nearly zero, the inverse is not available.

Reduced row echelon form is used to simplify matrices, solve systems of linear equations, and inspect pivot structure.

Yes. Use RREF mode and include the constants as the final column of Matrix A.

For matrix [[a, b], [c, d]], the determinant is ad - bc. A zero determinant means the 2x2 matrix does not have an inverse.

Each entry is a dot product: multiply matching row and column entries, then add them. For example, [1, 2] dot [3, 4] equals 1 x 3 + 2 x 4 = 11.

Matrix multiplication depends on row-by-column order. Even when both products are valid, the row and column pairings usually produce different entries.

A pivot is a leading nonzero entry used to eliminate other entries in its column. Pivot positions help show whether a system has one solution, no solution, or infinitely many solutions.

The page focuses on decimal output and readable step notes. For exact symbolic fractions, use a symbolic algebra tool or check small entries with a fraction calculator.

The matrix may be singular or nearly singular. A square matrix still needs nonzero determinant and stable row-reduction behavior to produce a reliable inverse.