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:

\[A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix}, B = \begin{bmatrix} 9 & 8 & 7 \\ 6 & 5 & 4 \\ 3 & 2 & 1 \end{bmatrix}\]

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:

\[C = AB = \begin{bmatrix} 30 & 24 & 18 \\ 84 & 69 & 54 \\ 138 & 114 & 90 \end{bmatrix}\]

Visual Representation

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