Matrix Addition Calculator
Add two or more matrices of the same dimensions. Enter your matrices below and get instant results with step-by-step solutions.
Matrix A
Matrix B
Result
How to Add Matrices
Same Dimensions
Matrices must have the same number of rows and columns to be added together.
Element-wise Addition
Add corresponding elements from each matrix to get the result matrix.
Commutative Property
Matrix addition is commutative: A + B = B + A
Associative Property
Matrix addition is associative: (A + B) + C = A + (B + C)
Mathematical Theory & History
What is Matrix Addition?
Matrix addition is a fundamental operation in linear algebra where corresponding elements of two matrices of the same dimensions are added together. The operation is defined as:
Formula: (A + B)ij = Aij + Bij
Where A and B are matrices of the same size, and i,j represent the row and column indices.
Historical Background
Matrix operations were formalized by British mathematician Arthur Cayley in the 1850s. The term "matrix" was introduced by James Joseph Sylvester in 1850, derived from the Latin word meaning "womb" or "breeding ground."
Cayley's 1858 paper "A Memoir on the Theory of Matrices" established the foundation of matrix algebra, including addition, multiplication, and other operations that are essential in modern mathematics, physics, and computer science.
Properties of Matrix Addition
Commutative Property
A + B = B + A
Associative Property
(A + B) + C = A + (B + C)
Identity Element
A + 0 = A (where 0 is the zero matrix)
Inverse Element
A + (-A) = 0
Real-World Applications
Computer Graphics
Combining transformation matrices for 3D rendering and animation
Economics
Adding input-output matrices to analyze economic systems
Engineering
Structural analysis and combining load matrices in civil engineering
Data Science
Combining datasets and feature matrices in machine learning
Frequently Asked Questions
No, matrix addition is only defined for matrices of the same dimensions. Both matrices must have the same number of rows and columns. If you try to add matrices of different sizes, the operation is undefined in linear algebra.
Adding a matrix to itself (A + A) is equivalent to multiplying the matrix by the scalar 2 (2A). Each element in the resulting matrix will be twice the value of the corresponding element in the original matrix.
Yes! Matrix addition is associative, so you can add multiple matrices: A + B + C + D. You can group them in any order: (A + B) + (C + D) or A + (B + C + D). All matrices must have the same dimensions.
The zero matrix (all elements are 0) acts as the additive identity. Adding a zero matrix to any matrix A results in A itself: A + 0 = A. This is similar to how adding 0 to any number leaves it unchanged.
Matrix addition works the same way with fractions and decimals. Simply add the corresponding elements using normal arithmetic rules. For fractions, find common denominators when necessary. Our calculator handles both fractions and decimals automatically.
Yes, matrix addition is exactly element-wise addition. Each element in the result matrix is the sum of the corresponding elements from the input matrices. This is different from matrix multiplication, which involves more complex calculations.
Common mistakes include: 1) Trying to add matrices of different sizes, 2) Confusing matrix addition with matrix multiplication, 3) Adding elements incorrectly (wrong positions), and 4) Forgetting that addition is commutative (A + B = B + A).