3x3 Matrix Multiplication

What is Matrix Multiplication?

Matrix multiplication is a fundamental operation in linear algebra. For two matrices A and B, their product AB is defined only if the number of columns in A equals the number of rows in B. In the case of 3x3 matrices, both matrices have 3 rows and 3 columns, making their multiplication always possible.

The Matrix Multiplication Formula

For two 3x3 matrices A and B, their product C = AB is defined as:

$$C_{ij}=\sum _{k=1}^3{A}_{ik}\times B_{kj}$$

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

Calculation Steps

To multiply two 3x3 matrices:

1. For each element in the resulting matrix:
   • Multiply corresponding elements from the row of A and column of B
   • Sum these products
2. Repeat this process for all 9 elements of the resulting matrix

Example Calculation

Let's multiply matrix A by matrix B:

$$A=\left[\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\\ 7& 8& 9\end{array}\right],B=\left[\begin{array}{ccc}9& 8& 7\\ 6& 5& 4\\ 3& 2& 1\end{array}\right]$$

Calculating the first element of C:

$$C_{11}=(1\times 9)+(2\times 6)+(3\times 3)=9+12+9=30$$

Continuing this process for all elements, we get:

$$C=AB=\left[\begin{array}{ccc}30& 24& 18\\ 84& 69& 54\\ 138& 114& 90\end{array}\right]$$

Visual Representation

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