Math & Statistics
Matrix Calculator – Add, Subtract, Multiply & Inverse Matrices Online
Use our free Matrix Calculator to perform matrix operations including addition, subtraction, multiplication, determinant, and inverse. Solve matrix problems quickly and accurately.
What is a Matrix Calculator?
A matrix is a rectangular array of numbers arranged in rows and columns, used extensively in linear algebra, physics, computer graphics, and data science. This calculator lets you perform key matrix operations including addition, subtraction, multiplication, determinant, transpose, and inverse.
Key rules: Addition and subtraction require same-sized matrices. For multiplication, the number of columns in Matrix A must equal the number of rows in Matrix B. Determinant and inverse only work on square matrices.
Matrix A
Matrix B
How to Use This Calculator
- Select the matrix operation you want to perform
- Set the matrix dimensions using the size dropdowns
- Enter values into the matrix cells
- Click Calculate to see the result
Common Use Cases
- Linear algebra homework
- Systems of equations
- Computer graphics
- Data transformations
- Engineering analysis
- Machine learning
Matrix Operations Reference
| Operation | Requirement | Result Size | Description |
|---|---|---|---|
| Addition | Same dimensions | Same as inputs | Add corresponding elements |
| Subtraction | Same dimensions | Same as inputs | Subtract corresponding elements |
| Multiplication | Cols(A) = Rows(B) | Rows(A) × Cols(B) | Dot product of rows and columns |
| Determinant | Square matrix | Scalar value | Measures matrix "volume scaling" |
| Transpose | Any matrix | Cols × Rows | Flip rows and columns |
| Inverse | Square, det ≠ 0 | Same as input | A × A⁻¹ = Identity matrix |
Common Special Matrices
| Type | Definition | Example (2×2) | Properties |
|---|---|---|---|
| Identity | 1s on diagonal, 0s elsewhere | [1,0; 0,1] | A × I = A |
| Zero | All elements are 0 | [0,0; 0,0] | A + 0 = A |
| Diagonal | Non-zero only on diagonal | [3,0; 0,5] | Easy to invert |
| Symmetric | A = Aᵀ | [1,2; 2,3] | Real eigenvalues |
| Singular | Determinant = 0 | [1,2; 2,4] | No inverse exists |
FAQ – Matrix Calculator
What does a Matrix Calculator do?
A Matrix Calculator performs mathematical operations on matrices, including addition, subtraction, multiplication, transposition, inversion, and determinant calculation. It helps simplify linear algebra problems efficiently.
What is a matrix?
A matrix is a rectangular array of numbers arranged in rows and columns. It is widely used in mathematics, physics, computer science, and data analysis to represent and solve linear equations, transformations, and datasets.
What is the determinant of a matrix?
The determinant is a special scalar value computed from a square matrix. It tells you whether the matrix is invertible (det ≠ 0) or singular (det = 0), and represents the scaling factor of the linear transformation the matrix represents.
How does matrix multiplication work?
Matrix multiplication involves multiplying rows of the first matrix by columns of the second matrix and summing the products. The operation is only valid if the number of columns in the first matrix equals the number of rows in the second. The result has dimensions Rows(A) × Cols(B).
What is the inverse of a matrix?
The inverse of a square matrix A is a matrix A⁻¹ such that A × A⁻¹ = Identity matrix. It only exists when the determinant is non-zero. The inverse is used to solve systems of linear equations: if Ax = b, then x = A⁻¹b.
What is the transpose of a matrix?
The transpose of a matrix is obtained by flipping it over its diagonal, converting rows into columns and vice versa. A matrix with dimensions m×n becomes n×m after transposition.
Why can't I add matrices of different sizes?
Matrix addition and subtraction are element-wise operations — each element in one matrix is added to the corresponding element in the other. This requires both matrices to have exactly the same number of rows and columns.
What does it mean when a matrix is singular?
A singular matrix has a determinant of zero, meaning it has no inverse. Geometrically, it collapses space into a lower dimension. For example, a 2×2 singular matrix maps all points onto a line or a single point.
What matrix sizes does this calculator support?
This calculator supports matrices up to 4×4 for all operations. For addition and subtraction, both matrices must be the same size. For multiplication, the column count of A must match the row count of B.
Who uses matrix calculations?
Matrices are used by students studying linear algebra, engineers for structural analysis, data scientists for machine learning, game developers for 3D graphics transformations, economists for input-output models, and physicists for quantum mechanics.