The matrix calculator performs addition, subtraction, multiplication, transpose, and determinant operations on 2×2 and 3×3 matrices. Enter values and select an operation.
Matrix Calculator
How to Use the Matrix Calculator
Matrices are rectangular arrays of numbers used to solve systems of equations, represent linear transformations, and model data in statistics and machine learning.
Matrix Multiplication Rules
For same-size square matrices, multiply each row of A by each column of B: result[i][j] = dot product of row i of A and column j of B. Matrix multiplication is NOT commutative: A×B ≠ B×A in general. This calculator shows A×B.
Determinant
2×2: det = ad − bc. 3×3: use cofactor expansion along first row. det = 0 means the matrix is singular (no inverse, system has no unique solution). det = ±1 means the matrix is "unimodular" — common in number theory.
Transpose
The transpose Aᵀ flips rows and columns: Aᵀ[i][j] = A[j][i]. A symmetric matrix satisfies A = Aᵀ. Used extensively in linear algebra, statistics (covariance matrices), and machine learning.
FAQ
What is matrix multiplication?
To multiply matrices A × B, the number of columns in A must equal the number of rows in B. The (i,j) entry of the result = dot product of row i of A with column j of B. Matrix multiplication is not commutative: A×B ≠ B×A in general. A 2×3 matrix × 3×2 matrix gives a 2×2 result.
What is a matrix determinant?
The determinant of a 2×2 matrix [[a,b],[c,d]] = ad − bc. For a 3×3 matrix, use cofactor expansion. The determinant = 0 means the matrix is singular (not invertible). |det| = area of the parallelogram formed by the row vectors (2D) or volume (3D).
Is this calculator free?
Yes, completely free with no signup required. All calculations run in your browser.
Is my data private?
Yes. All calculations run locally. Nothing is transmitted.
What is the inverse of a matrix?
The inverse A⁻¹ satisfies A × A⁻¹ = I (identity matrix). For 2×2: if A = [[a,b],[c,d]], then A⁻¹ = (1/det) × [[d,-b],[-c,a]]. A matrix has an inverse only if det(A) ≠ 0. Inverse matrices are used to solve systems of linear equations: Ax = b → x = A⁻¹b.