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Matrix Calculator — Free 2026

Add, subtract, multiply matrices, find determinants, transpose, and inverse for 2x2 and 3x3 matrices.

Matrix Size

Operation

Matrix A

Matrix B

Result

How It Works

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Enter matrix values
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Understanding Matrix Operations

Matrices are rectangular arrays of numbers that form the backbone of linear algebra. They represent systems of linear equations, geometric transformations, data in machine learning, and quantum states in physics. Every 3D video game, every Google search ranking, and every neural network relies on matrix operations at its core.

Basic Matrix Arithmetic

Matrix addition and subtraction are element-wise: you add or subtract corresponding entries. Both matrices must have the same dimensions. Matrix multiplication is more involved — the element at position (i,j) in the result is the dot product of row i from the first matrix and column j from the second. Unlike regular multiplication, matrix multiplication is not commutative: A × B generally does not equal B × A.

Determinants, Transpose, and Inverse

The determinant is a single number that captures key properties of a square matrix. A non-zero determinant means the matrix is invertible. The transpose swaps rows and columns. The inverse of matrix A (written A^-1) is the matrix such that A × A^-1 = I (the identity matrix). For related calculations, explore the scientific calculator or the fraction calculator for precise arithmetic with rational entries.

Real-World Applications

Computer graphics use 4x4 matrices for 3D transformations. Machine learning uses matrix multiplication in neural networks. Economists use input-output matrices to model economies. Engineers use them for stress analysis and circuit design. Understanding matrices unlocks powerful tools across virtually every quantitative field.

Frequently Asked Questions

What is a matrix?

A matrix is a rectangular array of numbers arranged in rows and columns. Matrices are used in linear algebra, physics, computer graphics, statistics, and engineering. A 2x2 matrix has 2 rows and 2 columns, while a 3x3 matrix has 3 rows and 3 columns.

How do you multiply two matrices?

To multiply matrices A and B, each element in the result is the dot product of a row from A and a column from B. For two 2x2 matrices, the element at row i, column j equals A[i][0]*B[0][j] + A[i][1]*B[1][j]. Matrix multiplication is not commutative — A times B does not always equal B times A.

What is a matrix determinant?

The determinant is a scalar value computed from a square matrix that encodes important properties. For a 2x2 matrix [[a,b],[c,d]], the determinant is ad - bc. A non-zero determinant means the matrix is invertible. The determinant also represents the scaling factor of the linear transformation described by the matrix.

When is a matrix not invertible?

A matrix is not invertible (also called singular) when its determinant equals zero. This means the matrix maps some non-zero vectors to zero, losing information. Geometrically, a singular matrix collapses space into a lower dimension — for example, squashing a plane into a line.