Matrix Multiplication

Implement matrix multiplication for two matrices A and B. This operation is crucial in linear algebra and computer vision, as it represents the composition of linear transformations.

Given matrices $A$ of size $m \times n$ and $B$ of size $n \times p$ , the product $C = A \times B$ is an $m \times p$ matrix.

Initialize the result matrix $C$ with zeros.
For each element $C_{ij}$ , compute the dot product of row $i$ from $A$ and column $j$ from $B$ . The resulting matrix $C$ represents the composition of the transformations represented by $A$ and $B$ .

C_{ij} = \sum_{k=1}^{n} A_{ik} \cdot B_{kj}

This technique is widely used in image processing and machine learning applications.

Example:

Input:

A = [[1, 2], [3, 4]]
B = [[5, 6], [7, 8]]

Output:

[[19, 22], [43, 50]]

Reasoning:

Step-by-step calculation using the matrix multiplication formula:

$C_{ij} = \sum_{k=1}^{n} A_{ik} \times B_{kj}$

Matrix $A$ (2×2) × Matrix $B$ (2×2) = Result $C$ (2×2)

$C_{00}$ : Row 0 of $A$ · Column 0 of $B$ $[1, 2] \cdot [5, 7] = (1 \times 5) + (2 \times 7) = 5 + 14 = 19$
$C_{01}$ : Row 0 of $A$ · Column 1 of $B$ $[1, 2] \cdot [6, 8] = (1 \times 6) + (2 \times 8) = 6 + 16 = 22$
$C_{10}$ : Row 1 of $A$ · Column 0 of $B$ $[3, 4] \cdot [5, 7] = (3 \times 5) + (4 \times 7) = 15 + 28 = 43$
$C_{11}$ : Row 1 of $A$ · Column 1 of $B$ $[3, 4] \cdot [6, 8] = (3 \times 6) + (4 \times 8) = 18 + 32 = 50$
Final result: $C = \begin{pmatrix} 19 & 22 \\ 43 & 50 \end{pmatrix}$

Constraints:

Matrix A has dimensions m×n, Matrix B has dimensions n×p
1 ≤ m, n, p ≤ 100
Elements are floating-point numbers
Round each element of the result to 4 decimal places

Example:

Constraints:

Test Results