Matrix Calculator

Calculate matrix addition, multiplication, and determinant free online for 2x2 and 3x3 matrices. Real linear algebra formulas, not simplified approximations.

Matrix A

Matrix B

Browser only -- real linear algebra calculations.

How to Use This Tool

Choose matrix size2x2 or 3x3

Enter values for both matricesEdit each cell directly

Choose an operationAddition, multiplication, or determinant

Matrix Calculator is a free online tool that works entirely in your browser — no signup, no data uploaded, no limits.

Real Linear Algebra, Not Simplified Shortcuts

Matrix operations follow specific, well-defined rules that differ meaningfully from regular number arithmetic -- multiplication combines rows and columns through sums of products, and determinant calculation uses cofactor expansion for matrices larger than 2x2. This calculator implements the actual mathematical procedures, not approximations.

Why Matrix Multiplication Order Matters

Unlike multiplying regular numbers, A×B and B×A generally produce different results for matrices -- a genuine mathematical property, not a quirk of this calculator, stemming from how the row-column combination process depends on which matrix's rows pair with which matrix's columns.

The Determinant's Practical Meaning

A zero determinant signals that a matrix cannot be inverted, with real consequences for solving systems of linear equations involving that matrix. The determinant also relates to how a matrix scales area (2x2) or volume (3x3) when applied as a geometric transformation.

Frequently Asked Questions

Matrix multiplication combines rows of the first matrix with columns of the second through a sum-of-products calculation for each resulting entry, which is why matrix dimensions must be compatible (columns of the first must match rows of the second) and why, unlike regular multiplication, the order genuinely matters -- A×B usually doesn't equal B×A.

The determinant indicates whether a matrix is invertible (a zero determinant means it isn't) and relates to how the matrix scales area or volume when used as a linear transformation -- a single number capturing important structural information about the matrix.

Using cofactor expansion along the first row, which breaks the 3x3 determinant down into a combination of three 2x2 determinants, each calculated using the simple ad-bc formula -- a real recursive structure, not a shortcut formula specific to 3x3.

Only matrices with identical dimensions can be added, since addition combines corresponding entries position by position -- mismatched dimensions have no defined corresponding entries to add.

Computer graphics transformations, systems of linear equations, data science and machine learning operations, and engineering simulations all rely on matrix operations as a fundamental computational tool.