Matrix Addition Calculator
Add two matrices of the same dimensions element by element.
Enter rows separated by semicolons, columns by commas. Example: 1,2;3,4
What Is Matrix Addition?
Matrix addition combines two matrices of identical dimensions by adding their corresponding entries. If A and B are both m×n matrices, their sum C = A + B is also an m×n matrix where:
Worked Example
Add the 2×2 matrices A and B:
[2, 4] [1, 7] [3, 11]
Each position is computed independently: top-left = 1+5 = 6, top-right = 3+2 = 5, etc.
Properties of Matrix Addition
- Commutative: A + B = B + A
- Associative: (A + B) + C = A + (B + C)
- Additive identity: A + O = A, where O is the zero matrix
- Additive inverse: A + (-A) = O, where -A has all signs flipped
Dimension Requirement
Matrix addition is only defined when both matrices have exactly the same number of rows AND the same number of columns. You cannot add a 2×3 matrix to a 3×2 matrix — dimensions must match exactly.
Real-World Applications
- Computer graphics: Adding transformation matrices to combine effects like translation and rotation.
- Economics: Combining supply and demand tables represented as matrices.
- Networks: Adding adjacency matrices of graphs to merge network connections.
- Statistics: Adding matrices of data from different time periods to get totals.
Frequently Asked Questions
When can two matrices be added?
Two matrices can be added only when they have the same number of rows and columns. Each element in the result is the sum of the corresponding elements from A and B.
Is matrix addition commutative?
Yes. A + B = B + A for any two matrices of the same dimensions. Matrix addition is both commutative and associative.
What is the identity element for matrix addition?
The zero matrix (all entries = 0) is the additive identity. A + O = A for any matrix A of matching dimensions.
How does matrix addition relate to scalar multiplication?
Together they form a vector space. The distributive law holds: k(A+B) = kA + kB and (k+m)A = kA + mA.