Add or subtract matrices up to 5×5 with element-wise breakdown, scalar combination mode, Frobenius norms, comparison heat map, and detailed result tables.
Matrix addition and subtraction are the simplest yet most fundamental operations in linear algebra. Two matrices can be added or subtracted only if they share the same dimensions — matching rows and columns — and the operation is performed element by element: each entry in the result is the sum (or difference) of the corresponding entries in the two input matrices.
This calculator handles square matrices up to 5×5 and supports three modes: standard addition (A + B), subtraction (A − B), and scalar combination (kA + lB), where you can weight each matrix by a scalar before combining them. The scalar combination mode is especially useful in linear interpolation, weighted averages, and blending transformations.
Beyond the result matrix, the calculator provides rich analysis including the Frobenius norm of each matrix (a measure of overall magnitude), element-wise difference heat maps that visually highlight where the matrices differ most, and a complete element-by-element breakdown table. These tools help you understand not just the result but the relationship between the input matrices.
Matrix addition is commutative (A + B = B + A) and associative ((A + B) + C = A + (B + C)). It distributes over scalar multiplication: k(A + B) = kA + kB. The zero matrix serves as the additive identity, and every matrix has an additive inverse (−A).
Even element-wise addition becomes error-prone for large matrices — a single misaligned index throws off a whole row. With scalar combination mode (kA + lB), tracking two scaling factors across every element multiplies the opportunity for mistakes. This calculator handles addition, subtraction, and weighted combination for matrices up to 5×5, showing a heat map of element-wise differences, Frobenius norms, and a full properties table. It is the quickest way to verify hand-computed results or explore how linear combinations of matrices behave.
Addition: (A + B)ᵢⱼ = aᵢⱼ + bᵢⱼ; Scalar combination: (kA + lB)ᵢⱼ = k·aᵢⱼ + l·bᵢⱼ; Frobenius norm: ‖A‖_F = √(Σᵢⱼ aᵢⱼ²)
Result: [[6,8],[10,12]]
Each element is the sum of corresponding elements: 1+5=6, 2+6=8, 3+7=10, 4+8=12.
Matrix addition and subtraction are defined element-wise: (A ± B)ᵢⱼ = aᵢⱼ ± bᵢⱼ. Both operands must have exactly the same dimensions — a 2×3 matrix cannot be added to a 3×2 matrix. This contrasts with matrix multiplication, which only requires inner dimensions to match. Addition is **commutative** (A + B = B + A) and **associative** ((A + B) + C = A + (B + C)), mirroring ordinary number addition. The zero matrix serves as the additive identity, and −A is the additive inverse.
Scalar combination computes kA + lB, weighting each matrix before summing. Setting k = l = 0.5 produces the element-wise average, useful for blending transformations in computer graphics. Setting k = 1 − t and l = t gives a linear interpolation (lerp) between two matrices, commonly used in animation and control systems. The Frobenius norm of the result satisfies the triangle inequality: ‖kA + lB‖_F ≤ |k|‖A‖_F + |l|‖B‖_F.
In **superposition** (physics and engineering), the response to combined inputs is the sum of individual responses — represented as matrix addition. In **image processing**, adding two image matrices blends them together, while subtraction reveals differences (change detection). In **finite element analysis**, global stiffness matrices are assembled by adding local element stiffness matrices. In **statistics**, covariance matrices of independent random vectors are additive. Understanding matrix addition is the first step toward mastering all matrix algebra.
No. Matrix addition requires both matrices to have exactly the same dimensions — same number of rows and same number of columns. There is no way to add a 2×3 matrix to a 3×2 matrix.
Scalar combination computes kA + lB, where k and l are scalars (numbers). This lets you weight each matrix differently before adding, useful for interpolation and blending.
The Frobenius norm ‖A‖_F is the square root of the sum of squares of all elements. It measures the overall "magnitude" of a matrix and is analogous to the Euclidean norm for vectors.
No. A − B ≠ B − A in general. However, A − B = −(B − A), so the results are negatives of each other.
The heat map displays |aᵢⱼ − bᵢⱼ| for each position, with darker red indicating larger differences. This helps you quickly spot where the two matrices diverge most.
The zero matrix (all entries are 0) of the same dimensions. Adding the zero matrix to any matrix A gives A back: A + 0 = A.