Matrix Addition
Add matrices entry by entry step by step · 행렬 덧셈
1. What Is Matrix Addition?
Matrix addition combines two matrices of the same size by adding their corresponding entries. The result is a new matrix with the same number of rows and columns. Unlike matrix multiplication, addition is straightforward: line the two grids up and add each pair of numbers that sit in the same position.
The one rule you must respect is dimension matching. You can add a 2×3 matrix to another 2×3 matrix, but you cannot add a 2×3 to a 3×2 — there is no entry to pair with, so the sum is undefined.
행렬 덧셈은 같은 크기의 두 행렬을 같은 위치의 원소끼리 더하는 연산입니다. 결과 행렬의 크기는 그대로 유지됩니다. 행과 열의 수가 정확히 같아야만 더할 수 있으며, 크기가 다르면 덧셈은 정의되지 않습니다.
2. The Element-Wise Rule
Each entry of the sum is the sum of the matching entries:
(A + B)ᵢⱼ = aᵢⱼ + bᵢⱼ
Worked example. For A = [[1, 2], [3, 4]] and B = [[5, 6], [7, 8]]:
A + B = [[1+5, 2+6], [3+7, 4+8]] = [[6, 8], [10, 12]]
합의 각 원소는 같은 위치 원소들의 합입니다: (A + B)ᵢⱼ = aᵢⱼ + bᵢⱼ. 예를 들어 [[1, 2], [3, 4]] + [[5, 6], [7, 8]] = [[6, 8], [10, 12]] 입니다.
3. A 3×3 Example
The same rule scales to any size. For two 3×3 matrices:
[[2, 0, 1], [3, 5, 4], [1, 1, 1]] + [[1, 2, 3], [0, 1, 2], [4, 4, 4]]
Add position by position to get:
[[3, 2, 4], [3, 6, 6], [5, 5, 5]]
Because addition is done independently in each cell, the order does not matter — adding B to A gives exactly the same result as adding A to B.
규칙은 크기와 무관하게 동일합니다. 위 3×3 예제의 합은 [[3, 2, 4], [3, 6, 6], [5, 5, 5]] 입니다. 각 칸이 독립적으로 더해지므로 더하는 순서는 결과에 영향을 주지 않습니다.
4. Properties of Matrix Addition
- Commutative. A + B = B + A. The order of the two matrices never changes the sum.
- Associative. (A + B) + C = A + (B + C), so you can add a chain of matrices in any grouping.
- Zero matrix is the identity. Adding the all-zero matrix 0 leaves a matrix unchanged: A + 0 = A.
- Transpose distributes. (A + B)ᵀ = Aᵀ + Bᵀ.
- Scalars distribute. c(A + B) = cA + cB for any number c.
These mirror the rules of ordinary number addition — which is exactly why addition is the gentle on-ramp to matrix algebra before the trickier multiplication.
행렬 덧셈은 교환법칙(A + B = B + A), 결합법칙((A + B) + C = A + (B + C))이 성립하고, 영행렬이 항등원(A + 0 = A)입니다. 또 (A + B)ᵀ = Aᵀ + Bᵀ, c(A + B) = cA + cB 가 성립합니다. 일반 수 덧셈과 같은 규칙을 따릅니다.
5. Frequently Asked Questions
Can you add matrices of different sizes? No. The two matrices must have identical dimensions — the same number of rows and the same number of columns. Otherwise the sum is undefined.
Is matrix addition commutative? Yes. A + B = B + A always holds, because each entry is just a sum of two numbers.
What is the zero matrix? A matrix whose entries are all 0. It is the additive identity: adding it to any matrix of the same size leaves that matrix unchanged.
크기가 다른 행렬은 더할 수 없습니다(행·열 수가 모두 같아야 함). 덧셈은 교환법칙이 성립합니다. 영행렬은 모든 원소가 0인 행렬로, 덧셈의 항등원입니다.
Ready to practice? Drill matrix addition and the rest of linear algebra on C:Matrix, or review the full matrix reference and the closely related matrix subtraction.