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:

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

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:

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

Visual Representation

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