Matrix Subtraction
Subtract matrices entry by entry step by step · 행렬 뺄셈
1. What Is Matrix Subtraction?
Matrix subtraction takes two matrices of the same size and subtracts the second from the first, entry by entry. Like addition, it works position by position and the result keeps the same dimensions. You can also think of A − B as adding the negative: A − B = A + (−B), where −B flips the sign of every entry of B.
As with addition, the matrices must have identical dimensions. Subtracting a 2×2 from a 3×3 is undefined because the entries do not line up.
행렬 뺄셈은 같은 크기의 두 행렬에서 같은 위치 원소끼리 빼는 연산입니다. A − B = A + (−B)로 볼 수도 있는데, 여기서 −B는 B의 모든 원소 부호를 바꾼 행렬입니다. 덧셈과 마찬가지로 두 행렬의 크기가 같아야 합니다.
2. The Element-Wise Rule
Each entry of the difference is the difference of the matching entries:
(A − B)ᵢⱼ = aᵢⱼ − bᵢⱼ
Worked example. For A = [[9, 8], [7, 6]] and B = [[1, 2], [3, 4]]:
A − B = [[9−1, 8−2], [7−3, 6−4]] = [[8, 6], [4, 2]]
차의 각 원소는 같은 위치 원소들의 차입니다: (A − B)ᵢⱼ = aᵢⱼ − bᵢⱼ. 예를 들어 [[9, 8], [7, 6]] − [[1, 2], [3, 4]] = [[8, 6], [4, 2]] 입니다.
3. Order Matters
Unlike addition, subtraction is not commutative: A − B is generally not equal to B − A. In fact, swapping the order flips the sign of every entry:
B − A = −(A − B)
Using the example above, B − A = [[−8, −6], [−4, −2]] — the same numbers with opposite signs. The only matrix that comes back to zero is a matrix minus itself: A − A = 0, the zero matrix.
덧셈과 달리 뺄셈은 교환법칙이 성립하지 않습니다: 일반적으로 A − B ≠ B − A 이고, 순서를 바꾸면 모든 원소의 부호가 반대가 됩니다(B − A = −(A − B)). 자기 자신을 빼면 A − A = 0(영행렬)이 됩니다.
4. Properties of Matrix Subtraction
- Not commutative. A − B ≠ B − A in general; order changes the sign.
- Self-difference is zero. A − A = 0.
- As addition of a negative. A − B = A + (−B), so every subtraction is really an addition in disguise.
- Transpose distributes. (A − B)ᵀ = Aᵀ − Bᵀ.
- Same-size requirement. Defined only when A and B share dimensions.
Because subtraction is addition of a negative, all the convenience of addition still applies — you just have to watch the order.
뺄셈은 교환법칙이 성립하지 않고(A − B ≠ B − A), A − A = 0, A − B = A + (−B), (A − B)ᵀ = Aᵀ − Bᵀ 가 성립하며, 크기가 같을 때만 정의됩니다. 순서에만 주의하면 덧셈과 같은 방식으로 다룰 수 있습니다.
5. Frequently Asked Questions
Does the order matter in matrix subtraction? Yes. A − B and B − A differ by a sign on every entry, so subtraction is not commutative.
Can you subtract matrices of different sizes? No. The two matrices must have the same number of rows and columns.
What is A − A? The zero matrix — every entry cancels to 0.
뺄셈은 순서가 중요합니다(A − B ≠ B − A). 크기가 다른 행렬은 뺄 수 없습니다. A − A는 영행렬입니다.
Ready to practice? Drill matrix subtraction and the rest of linear algebra on C:Matrix, or review the full matrix reference and the closely related matrix addition.