Error converting content: marked is not a function
title:: Matrix Multiplication: A Choreographed Dance
**1. Concept Overview:**
- **Matrices**: Rectangular arrays of numbers or expressions.
- **Matrix Multiplication**: Combining two matrices into a new matrix.
- **Dimensions**: For \( A \) (of size \( m \times n \)) and \( B \) (of size \( n \times p \)), the result \( C \) is \( m \times p \).
- **2. ASCII Visualization:**
```
Matrix A (m x n) Matrix B (n x p) Matrix C (m x p)
[ a11 a12 ... a1n ] [ b11 b12 ... b1p ] [ c11 c12 ... c1p ]
[ a21 a22 ... a2n ] [ b21 b22 ... b2p ] = [ c21 c22 ... c2p ]
[ . . ... . ] [ . . ... . ] [ . . ... . ]
[ am1 am2 ... amn ] [ bn1 bn2 ... bnp ] [ cm1 cm2 ... cmp ]
```
**3. Mathematical Representation:**
- **Equation**: \( c_{ij} = a_{i1} \cdot b_{1j} + a_{i2} \cdot b_{2j} + \ldots + a_{in} \cdot b_{nj} \).
- **4. Concrete Example:**
- **Matrices \( A \) and \( B \)**:
```
A = [[2, 3, 4], [5, 6, 7]]
B = [[8, 9], [10, 11], [12, 13]]
```
- **Raw Python Code**:
```python
def matrix_multiplication(A, B):
rows_A, cols_A = len(A), len(A[0])
rows_B, cols_B = len(B), len(B[0])
if cols_A != rows_B:
raise ValueError("Number of columns in A must be equal to the number of rows in B.")
C = [[0 for _ in range(cols_B)] for _ in range(rows_A)]
for i in range(rows_A):
for j in range(cols_B):
for k in range(cols_A):
C[i][j] += A[i][k] * B[k][j]
return C
C = matrix_multiplication(A, B)
```
- **5. Insights and Connections:**
- **Depth**: More than arithmetic; it's a geometric transformation.
- **Applications**: In computer graphics, data science, AI, etc.