3x3 Matrix Addition

What is Matrix Addition?

Matrix addition is a fundamental operation in linear algebra. It involves adding two matrices of the same dimensions element by element. For 3x3 matrices, we add corresponding elements from each matrix to create a new 3x3 matrix.

The Matrix Addition Formula

For two 3x3 matrices A and B, their sum C = A + B is calculated as:

\[C_{ij} = A_{ij} + B_{ij}\]

Where:

• \(C_{ij}\) is the element in the i-th row and j-th column of the resulting matrix C
• \(A_{ij}\) is the element in the i-th row and j-th column of matrix A
• \(B_{ij}\) is the element in the i-th row and j-th column of matrix B

Calculation Steps for 3x3 Matrix Addition

To add two 3x3 matrices A and B:

1. Ensure both matrices have the same dimensions (3x3 in this case)
2. Add corresponding elements from A and B
3. Place the result in the corresponding position of the new matrix C

Example Calculation

Let's add these two 3x3 matrices:

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

Adding A and B:

\[C = A + B = \begin{bmatrix} 1+9 & 2+8 & 3+7 \\ 4+6 & 5+5 & 6+4 \\ 7+3 & 8+2 & 9+1 \end{bmatrix} = \begin{bmatrix} 10 & 10 & 10 \\ 10 & 10 & 10 \\ 10 & 10 & 10 \end{bmatrix}\]

Visual Representation

This diagram visually represents the addition of matrices A and B to produce the result matrix.