Matrix Calculator
Perform matrix operations with ease, addition, subtraction, multiplication, determinant, inverse, and transpose. Enter matrices of any size up to 6×6 and get instant results. Perfect for students, engineers, and professionals.
What Is a Matrix?
A matrix is a rectangular array of numbers arranged in rows and columns. Matrices are fundamental in linear algebra and are used extensively in mathematics, physics, engineering, computer science, and economics. They provide a compact way to represent and manipulate linear transformations, systems of equations, and data sets.
This calculator supports the most common matrix operations:
- Addition: Add two matrices of the same dimensions (A + B).
- Subtraction: Subtract one matrix from another (A − B).
- Multiplication: Multiply two matrices (A × B) — requires the number of columns in A to equal the number of rows in B.
- Determinant: Calculate the determinant of a square matrix.
- Inverse: Find the inverse of an invertible square matrix.
- Transpose: Flip a matrix over its diagonal (swap rows and columns).
How the Matrix Calculator Works
The calculator uses standard linear algebra algorithms implemented in JavaScript:
Matrix Addition/Subtraction: Cij = Aij ± Bij (element-wise)
Matrix Multiplication: Cij = Σ(Aik × Bkj)
Determinant: Calculated using cofactor expansion for 2×2 and 3×3 matrices, and row reduction for larger matrices.
Inverse: Computed using the adjugate method or Gauss-Jordan elimination.
Transpose: Cij = Aji (swap rows and columns)
Simply select an operation, enter your matrix values, and click "Calculate" to get instant results.
Why Use This Matrix Calculator?
- Six Operations: Addition, subtraction, multiplication, determinant, inverse, and transpose.
- Flexible Sizes: Supports matrices from 1×1 up to 6×6.
- Visual Input: Intuitive grid-based input for easy data entry.
- Step-by-Step: Optional calculation steps for learning and verification.
- Free & Private: No registration, no data storage.
❓ Matrix Calculator FAQ
What is a matrix?
A matrix is a rectangular array of numbers arranged in rows and columns. It is a fundamental concept in linear algebra used to represent linear transformations, systems of equations, and data sets.
What operations does this matrix calculator support?
This calculator supports addition, subtraction, multiplication, determinant calculation, inverse finding, and transpose of matrices up to 6×6.
How do I add two matrices?
To add two matrices, they must have the same dimensions. Add corresponding elements: (A + B)ij = Aij + Bij. Select "Addition" as the operation and enter both matrices.
How do I multiply two matrices?
To multiply two matrices, the number of columns in matrix A must equal the number of rows in matrix B. The result has dimensions rows(A) × cols(B). Element Cij = Σ(Aik × Bkj).
What is the determinant of a matrix?
The determinant is a scalar value that can be computed from the elements of a square matrix. It provides important information about the matrix, such as whether it is invertible (determinant ≠ 0) and the volume scaling factor of the linear transformation.
What is the inverse of a matrix?
The inverse of a square matrix A is a matrix A⁻¹ such that A × A⁻¹ = I (the identity matrix). Only matrices with a non-zero determinant have an inverse. The inverse is used to solve systems of linear equations.
What is the transpose of a matrix?
The transpose of a matrix is obtained by swapping its rows and columns. If A is an m×n matrix, its transpose Aᵀ is an n×m matrix where (Aᵀ)ij = Aji.
What are the requirements for matrix multiplication?
For matrix multiplication A × B, the number of columns in A must equal the number of rows in B. If A is m×n and B is n×p, the result is m×p.
How do I calculate the determinant of a 3×3 matrix?
For a 3×3 matrix, the determinant is calculated using cofactor expansion: det(A) = a₁₁(a₂₂a₃₃ − a₂₃a₃₂) − a₁₂(a₂₁a₃₃ − a₂₃a₃₁) + a₁₃(a₂₁a₃₂ − a₂₂a₃₁).
What size matrices can I use?
This calculator supports matrices from 1×1 up to 6×6 for all operations, with the exception of determinant and inverse which require square matrices.
What is the identity matrix?
The identity matrix is a square matrix with 1s on the main diagonal and 0s elsewhere. It is the multiplicative identity for matrices: A × I = I × A = A.
What does it mean if a matrix is singular?
A singular matrix is a square matrix that does not have an inverse. This happens when the determinant is zero. Such matrices represent linear transformations that collapse space.
How do I use this calculator for systems of linear equations?
You can represent a system of linear equations as A×X = B. To solve for X, you can use the inverse: X = A⁻¹ × B. Use this calculator to find the inverse of the coefficient matrix A.
What is the rank of a matrix?
The rank of a matrix is the maximum number of linearly independent rows or columns. It indicates the dimension of the vector space spanned by its rows or columns. This calculator does not compute rank directly but it's closely related to the determinant and invertibility.
How accurate are the results?
The calculator uses double-precision floating-point arithmetic and provides results with up to 5 decimal places. For most practical purposes, the results are highly accurate. Rounding errors may occur for ill-conditioned matrices.
Can I use this calculator for complex numbers?
No, this calculator currently supports only real numbers. For complex matrix operations, you would need a specialized tool.
What is the difference between a matrix and a determinant?
A matrix is a rectangular array of numbers. A determinant is a scalar value that can be computed from a square matrix. The determinant is a property of the matrix, not a matrix itself.
How do I clear all matrix values?
Click the "Reset" button to clear all input fields and results. You can also manually update the matrix dimensions and re-enter values.
Is this matrix calculator free to use?
Yes, this calculator is completely free to use. No registration or personal data storage is required. All calculations are performed in your browser.