4x4 Matrix Multiplication

What is Matrix Multiplication?

Matrix multiplication is a fundamental operation in linear algebra. It involves multiplying two matrices to produce a new matrix. For 4x4 matrices, we multiply a 4x4 matrix by another 4x4 matrix to get a resulting 4x4 matrix.

The Matrix Multiplication Formula

For two 4x4 matrices A and B, their product C = AB is defined as:

\[C_{ij} = \sum_{k=1}^4 A_{ik} \cdot B_{kj}\]

Where:

• \(C_{ij}\) is the element in the i-th row and j-th column of the resulting matrix C
• \(A_{ik}\) is the element in the i-th row and k-th column of matrix A
• \(B_{kj}\) is the element in the k-th row and j-th column of matrix B
• The sum is taken over k from 1 to 4

Calculation Steps

To multiply two 4x4 matrices:

1. For each element in the resulting matrix:
2. Multiply corresponding elements from the row of A and column of B
3. Sum these products
4. Place the sum in the corresponding position of the resulting matrix C
5. Repeat this process for all 16 elements of the resulting matrix

Example Calculation

Let's multiply matrix A by matrix B:

\[A = \begin{bmatrix} 1 & 2 & 3 & 4 \\ 5 & 6 & 7 & 8 \\ 9 & 10 & 11 & 12 \\ 13 & 14 & 15 & 16 \end{bmatrix}, B = \begin{bmatrix} 17 & 18 & 19 & 20 \\ 21 & 22 & 23 & 24 \\ 25 & 26 & 27 & 28 \\ 29 & 30 & 31 & 32 \end{bmatrix}\]

Calculating the first element of C:

\[C_{11} = (1 \cdot 17) + (2 \cdot 21) + (3 \cdot 25) + (4 \cdot 29) = 250\]

Continuing this process for all elements, we get:

\[C = AB = \begin{bmatrix} 250 & 260 & 270 & 280 \\ 618 & 644 & 670 & 696 \\ 986 & 1028 & 1070 & 1112 \\ 1354 & 1412 & 1470 & 1528 \end{bmatrix}\]

Visual Representation

This diagram visually represents the multiplication of matrix A and matrix B, resulting in AB.