Matrix Multiplication

Multiply matrices row by column step by step · 행렬 곱셈

1. What Is Matrix Multiplication?

행렬 곱셈이란?

Matrix multiplication combines two matrices using a row-by-column rule, not by multiplying entries in the same position. To find the entry in row i, column j of the product, you take the dot product of row i of the first matrix with column j of the second: multiply the paired numbers and add them up.

This is the operation behind composing transformations, solving systems, and almost all of linear algebra — which is why it behaves so differently from the gentle, element-wise addition. The key requirement: the number of columns of the first matrix must equal the number of rows of the second.

행렬 곱셈은 같은 위치 원소를 곱하는 것이 아니라, "행 × 열" 규칙으로 계산합니다. 곱의 (i, j) 원소는 첫 행렬의 i행과 둘째 행렬의 j열의 내적(짝지은 수를 곱해 합)입니다. 첫 행렬의 열 수와 둘째 행렬의 행 수가 같아야 곱할 수 있습니다.

2. The Row-by-Column Rule

행 × 열 규칙

For the product C = AB, each entry is a sum of products:

cᵢⱼ = Σₖ aᵢₖ · bₖⱼ

The product C has as many rows as A and as many columns as B.

곱 C = AB의 각 원소는 cᵢⱼ = Σₖ aᵢₖ · bₖⱼ 입니다. 결과 C의 행 수는 A와 같고, 열 수는 B와 같습니다.

3. Worked 2×2 Example

2×2 풀이 예제

For A = [[1, 2], [3, 4]] and B = [[5, 6], [7, 8]], compute each entry as row · column:

c₁₁ = 1·5 + 2·7 = 5 + 14 = 19
c₁₂ = 1·6 + 2·8 = 6 + 16 = 22
c₂₁ = 3·5 + 4·7 = 15 + 28 = 43
c₂₂ = 3·6 + 4·8 = 18 + 32 = 50

AB = [[19, 22], [43, 50]]

If you reverse the order, BA = [[23, 34], [31, 46]] — a different matrix. That single fact, AB ≠ BA, is the most important thing to remember about matrix multiplication.

A = [[1, 2], [3, 4]], B = [[5, 6], [7, 8]]이면 AB = [[19, 22], [43, 50]] 입니다. 순서를 바꾼 BA = [[23, 34], [31, 46]]로 서로 다릅니다 — 즉 AB ≠ BA 입니다.

4. Properties of Matrix Multiplication

행렬 곱셈의 성질

Not commutative. AB ≠ BA in general; order matters.
Associative. (AB)C = A(BC), so you can regroup a chain of products.
Distributive. A(B + C) = AB + AC.
Identity matrix. The identity I (1s on the diagonal, 0s elsewhere) satisfies AI = IA = A.
Transpose reverses order. (AB)ᵀ = BᵀAᵀ.
Not element-wise. Multiplying matching entries instead is the Hadamard product — a different operation.

For two square matrices the determinant is multiplicative: det(AB) = det(A)·det(B).

행렬 곱셈은 교환법칙이 성립하지 않지만(AB ≠ BA), 결합법칙((AB)C = A(BC))과 분배법칙(A(B + C) = AB + AC)은 성립합니다. 항등행렬 I는 AI = IA = A, (AB)ᵀ = BᵀAᵀ, det(AB) = det(A)det(B)가 성립합니다. 같은 위치끼리 곱하는 것은 다른 연산(아다마르 곱)입니다.

5. Frequently Asked Questions

자주 묻는 질문

Is matrix multiplication commutative? No. AB and BA are usually different matrices, and BA may not even be defined. Always preserve the order.

Is it the same as multiplying matching entries? No. Entry-by-entry multiplication is the Hadamard product. Standard matrix multiplication uses the row-by-column dot product.

When can two matrices be multiplied? When the first matrix’s column count equals the second’s row count — an (m×n) times an (n×p) gives an (m×p) result.

행렬 곱셈은 교환법칙이 성립하지 않습니다(AB ≠ BA). 같은 위치끼리 곱하는 것(아다마르 곱)과 다릅니다. 첫 행렬의 열 수 = 둘째 행렬의 행 수일 때 곱할 수 있습니다((m×n)(n×p) = m×p).

Ready to practice? Drill matrix multiplication and the rest of linear algebra on C:Matrix, or review the full matrix reference and the related rectangular multiplication and inverse matrix.

실전 연습은 C:Matrix에서, 전체 개념은 행렬 레퍼런스와 직사각 행렬 곱셈, 역행렬 문서에서 이어집니다.

Practice matrix problems → C:Matrix