Perform matrix operations including addition, subtraction, multiplication, transpose, determinant, inverse, and scalar multiplication for matrices up to 5×5.
Matrices are fundamental structures in linear algebra that organize numbers into rows and columns, enabling powerful mathematical operations used throughout science, engineering, and computer graphics. Whether you're solving systems of linear equations, performing transformations in 3D space, or analyzing data in machine learning, matrix operations are indispensable tools.
This comprehensive matrix calculator supports all major operations for matrices up to 5×5: addition, subtraction, multiplication, transpose, determinant, inverse, and scalar multiplication. Simply enter your matrix values, choose an operation, and get instant results with detailed properties.
Understanding matrix properties like rank, trace, determinant, and sparsity helps you characterize the matrix and determine which operations are valid. A square matrix with a non-zero determinant is invertible, while the rank tells you the number of linearly independent rows or columns. The trace — the sum of diagonal elements — appears in many formulas across physics and statistics.
Use the preset buttons to explore common examples, or enter your own values. The calculator validates dimension compatibility for operations like multiplication (where the number of columns in A must equal the number of rows in B) and checks for singularity before attempting inversion.
Matrix operations involve tracking many indices and sign rules simultaneously — multiplication alone requires n³ individual multiply-add operations for an n×n matrix. Even simple operations like verifying that AB ≠ BA or checking that A·A⁻¹ = I are cumbersome by hand. This all-in-one calculator handles addition, subtraction, multiplication, transpose, determinant, inverse, and scalar multiplication for matrices up to 5×5, showing properties like rank, trace, and sparsity. It is the go-to tool for students doing homework, engineers verifying hand calculations, or anyone exploring matrix algebra interactively.
Matrix multiplication: (AB)ᵢⱼ = Σₖ aᵢₖ bₖⱼ; Determinant (2×2): ad − bc; Inverse: A⁻¹ = adj(A) / det(A)
Result: Determinant = 5
For the 2×2 matrix [[2,1],[1,3]], det = 2×3 − 1×1 = 5. Since det ≠ 0, the matrix is invertible.
**Addition and subtraction** require same-dimension matrices and operate element-wise: (A±B)ᵢⱼ = aᵢⱼ ± bᵢⱼ. **Scalar multiplication** scales every entry: (cA)ᵢⱼ = c·aᵢⱼ. **Matrix multiplication** is the most powerful: (AB)ᵢⱼ = Σₖ aᵢₖbₖⱼ, combining rows of A with columns of B. Multiplication is associative and distributive but **not commutative** — AB and BA are generally different or may not even be defined. The transpose (Aᵀ)ᵢⱼ = aⱼᵢ swaps rows and columns, with the important rule (AB)ᵀ = BᵀAᵀ.
The **determinant** is a scalar that encodes whether a matrix is invertible: det(A) ≠ 0 means A has a unique inverse A⁻¹ such that AA⁻¹ = I. The inverse is computed via the adjugate formula A⁻¹ = adj(A)/det(A) or by row-reducing [A|I]. The **rank** is the number of linearly independent rows (or columns) and determines the dimension of the column space. A full-rank n×n matrix is invertible; a rank-deficient matrix is singular.
In **computer graphics**, 4×4 matrices encode rotations, translations, and projections that transform 3D objects onto a 2D screen. In **physics**, matrices represent quantum states (density matrices), moments of inertia (inertia tensors), and stress/strain (elasticity tensors). In **data science**, the data matrix X has observations as rows and features as columns — matrix operations like XᵀX appear in regression, PCA, and covariance estimation. Understanding matrix operations is the prerequisite for nearly every topic in applied mathematics and computational science.
A matrix is a rectangular array of numbers arranged in rows and columns. An m×n matrix has m rows and n columns. Matrices are used to represent linear transformations, systems of equations, and data structures.
You can multiply matrix A (m×n) by matrix B (p×q) only if n = p — the number of columns in A must equal the number of rows in B. The result will be an m×q matrix.
The determinant is a scalar value that indicates whether a square matrix is invertible (det ≠ 0) or singular (det = 0). It also represents the scaling factor of the linear transformation defined by the matrix.
The inverse is computed using the adjugate method: A⁻¹ = adj(A) / det(A), where adj(A) is the transpose of the cofactor matrix. This only works when det(A) ≠ 0.
The rank is the maximum number of linearly independent rows (or columns). It equals the number of pivot positions in the row echelon form and determines the dimension of the column space.
The trace is the sum of the diagonal elements of a square matrix: tr(A) = a₁₁ + a₂₂ + ... + aₙₙ. It equals the sum of the eigenvalues and is invariant under similarity transformations.