Matrix Calculator

What is a matrix

A matrix is a rectangular array of numbers arranged in rows and columns. Matrices are widely used in mathematics, physics, computer graphics, machine learning, and economics to compactly represent systems of equations, transformations, and data.

Basic matrix operations

• Addition/subtraction: only possible for matrices of the same size — corresponding elements are simply added or subtracted.
• Multiplication: the number of columns in the first matrix must equal the number of rows in the second. The result is a new matrix with the rows of the first and columns of the second.
• Transpose: turns rows into columns — the element at (i,j) moves to position (j,i).
• Determinant: a number that can only be computed for square matrices, showing the scaling factor of the transformation and whether an inverse matrix exists.

The matrix multiplication rule

Unlike ordinary numbers, matrix multiplication is not commutative — meaning A×B usually doesn't equal B×A. Additionally, multiplication is only possible if the matrix dimensions are compatible: the number of columns in A must exactly match the number of rows in B.

Where matrices are used

• Computer graphics: rotating, scaling, and translating 3D objects.
• Machine learning: neural networks operate on data as matrices and tensors.
• Systems of equations: a compact way to write and solve systems of linear equations.
• Economics: input-output models, market equilibrium analysis.

This calculator supports matrices up to 4×4. For larger matrices or more complex operations (matrix inverse, eigenvalues), specialized software (MATLAB, Python NumPy) is recommended.