Matrix Addition Calculator

Add two matrices of the same dimensions

Matrix Addition Calculator – Add Two Matrices

Add two matrices of the same dimensions with our free online matrix addition calculator. Enter values via textarea or row-by-row input with step-by-step results.

Matrix A Size:

or custom×

Enter Matrix A values (each row on a new line, values separated by spaces or commas)

Current Matrix A: 2×2

Matrix B Size:

or custom×

Enter Matrix B values (each row on a new line, values separated by spaces or commas)

Current Matrix B: 2×2

What is Matrix Addition?

Matrix addition is the operation of adding two matrices of the same dimensions by adding their corresponding elements. If matrix A and matrix B both have dimensions m×n, their sum C = A + B also has dimensions m×n, where each element cᵢⱼ = aᵢⱼ + bᵢⱼ.

Matrix addition is only defined for matrices of the same size. You cannot add matrices with different dimensions.

Matrix Addition Formula

General Formula

If A = [aᵢⱼ] and B = [bᵢⱼ], then C = A + B = [aᵢⱼ + bᵢⱼ]

Each element in the result is the sum of corresponding elements from the input matrices

2×2 Example

[a b]   [e f]   [a+e  b+f]
[c d] + [g h] = [c+g  d+h]

3×3 Example

[a b c]   [j k l]   [a+j  b+k  c+l]
[d e f] + [m n o] = [d+m  e+n  f+o]
[g h i]   [p q r]   [g+p  h+q  i+r]

Properties of Matrix Addition

Key Properties

• Commutative: A + B = B + A
• Associative: (A + B) + C = A + (B + C)
• Identity: A + 0 = A (zero matrix)
• Inverse: A + (-A) = 0

Requirements

• Both matrices must have same dimensions
• Same number of rows
• Same number of columns
• Result has the same dimensions

Frequently Asked Questions

What happens if matrices have different sizes?

Matrix addition is only defined for matrices of the same dimensions. If A is 2×3 and B is 3×2, they cannot be added. The calculator will show an error message indicating the dimension mismatch.

Can I add more than two matrices?

Yes. Matrix addition is associative, so you can add multiple matrices: A + B + C = (A + B) + C. All matrices must have the same dimensions.

What is the zero matrix?

A zero matrix has all elements equal to zero. Adding a zero matrix to any matrix A gives A unchanged. It's the additive identity for matrices.

Is matrix addition commutative?

Yes. A + B = B + A for any matrices of the same size. The order doesn't matter because you're just adding numbers at each position.

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