3x3 Matrix Subtraction

What is Matrix Subtraction?

Matrix subtraction is a fundamental operation in linear algebra. For two matrices of the same size, subtraction is performed by subtracting corresponding elements. In the case of 3x3 matrices, we subtract each element of the second matrix from the corresponding element of the first matrix.

The Matrix Subtraction Formula

For two 3x3 matrices A and B, their difference A - B is defined as:

\[A - B = \begin{bmatrix} a_{11} - b_{11} & a_{12} - b_{12} & a_{13} - b_{13} \\ a_{21} - b_{21} & a_{22} - b_{22} & a_{23} - b_{23} \\ a_{31} - b_{31} & a_{32} - b_{32} & a_{33} - b_{33} \end{bmatrix}\]

Where:

• \(a_{ij}\) represents the element in the i-th row and j-th column of matrix A
• \(b_{ij}\) represents the element in the i-th row and j-th column of matrix B

Calculation Steps

To subtract two 3x3 matrices:

1. Ensure both matrices are 3x3
2. Subtract corresponding elements:
   • Subtract the element in the first row, first column of B from the same position in A
   • Continue this process for all 9 elements
3. The resulting matrix will also be 3x3

Example Calculation

Let's subtract matrix B from matrix A:

\[A = \begin{bmatrix} 5 & 2 & 1 \\ 3 & 4 & 6 \\ 1 & 7 & 2 \end{bmatrix}, B = \begin{bmatrix} 1 & 1 & 0 \\ 2 & 3 & 1 \\ 0 & 2 & 1 \end{bmatrix}\] \[A - B = \begin{bmatrix} 5-1 & 2-1 & 1-0 \\ 3-2 & 4-3 & 6-1 \\ 1-0 & 7-2 & 2-1 \end{bmatrix} = \begin{bmatrix} 4 & 1 & 1 \\ 1 & 1 & 5 \\ 1 & 5 & 1 \end{bmatrix}\]

Visual Representation

This diagram visually represents the subtraction of matrix B from matrix A, resulting in A - B.