🔲 Operations Calculators

Perform matrix arithmetic: addition, multiplication, scalar operations, transpose, and powers.

All Operations Tools

Matrix Addition Calculator Add two matrices of the same dimensions element by element. Matrix Multiplication Calculator Multiply two matrices where columns of A equal rows of B. Scalar Multiplication Calculator Multiply every element of a matrix by a scalar constant. Matrix Transpose Calculator Transpose a matrix by swapping its rows and columns. Matrix Power Calculator Raise a square matrix to an integer power by repeated multiplication.

Matrix Addition and Subtraction

Matrix addition and subtraction are element-wise operations that require both matrices to have identical dimensions (same number of rows and columns). Each element in the result matrix is the sum (or difference) of the corresponding elements in the two input matrices. For two 2×2 matrices A and B: (A+B)ᵢⱼ = Aᵢⱼ + Bᵢⱼ for all i,j. Matrix addition is commutative (A+B = B+A) and associative ((A+B)+C = A+(B+C)). These properties mirror ordinary arithmetic. Matrix subtraction is neither commutative nor associative. Addition and subtraction are used extensively in linear systems, computer graphics transformations, and neural network weight updates.

Matrix Multiplication

Matrix multiplication is not element-wise — it requires the number of columns in the first matrix to equal the number of rows in the second. If A is m×n and B is n×p, the product AB is m×p. Each element (AB)ᵢⱼ is the dot product of row i of A with column j of B: (AB)ᵢⱼ = Σₖ Aᵢₖ × Bₖⱼ. Matrix multiplication is associative and distributive but not commutative — AB ≠ BA in general, and BA may not even be defined if the dimensions don't permit it. Scalar multiplication multiplies every element by a constant k: (kA)ᵢⱼ = k × Aᵢⱼ. This is always defined and distributes over matrix addition: k(A+B) = kA + kB.

Matrix Transpose

The transpose of a matrix A, written Aᵀ, is formed by swapping rows and columns: (Aᵀ)ᵢⱼ = Aⱼᵢ. A 3×2 matrix transposes to a 2×3 matrix. Key properties: (Aᵀ)ᵀ = A (double transpose returns original); (A+B)ᵀ = Aᵀ+Bᵀ; (AB)ᵀ = BᵀAᵀ (note the reversal of order). A symmetric matrix satisfies A = Aᵀ — covariance matrices in statistics and many matrices in physics are symmetric. An orthogonal matrix satisfies AᵀA = I (identity), meaning Aᵀ = A⁻¹ — rotation matrices in 3D graphics are orthogonal, making their inverse trivially easy to compute as just the transpose.

Matrix Powers

A matrix power Aⁿ means multiplying the matrix A by itself n times: A² = A×A, A³ = A×A×A. Only square matrices (same number of rows and columns) can be raised to a power. A⁰ = I (the identity matrix, analogous to x⁰ = 1). Matrix powers appear in Markov chain analysis (Aⁿ gives transition probabilities after n steps), graph theory (Aⁿᵢⱼ counts the number of paths of length n from vertex i to vertex j), and population dynamics models. Computing large matrix powers directly is computationally expensive; the exponentiation by squaring algorithm computes Aⁿ in O(log n) multiplications by exploiting A⁸ = (A⁴)² = ((A²)²)².