Matrix Operations

What are Matrix Operations?

Matrix operations are fundamental mathematical procedures performed on matrices. They include addition, subtraction, multiplication, and more complex operations like finding inverses or determinants. These operations are crucial in various fields, including physics, computer graphics, and data analysis.

Key Matrix Operations

1. Matrix Addition and Subtraction

For matrices A and B of the same size:

\((A \pm B)_{ij} = A_{ij} \pm B_{ij}\)

Each element in the resulting matrix is the sum or difference of the corresponding elements in A and B.

2. Matrix Multiplication

For matrices A (m×n) and B (n×p):

\((AB)_{ij} = \sum_{k=1}^n A_{ik}B_{kj}\)

Each element in the resulting matrix is the dot product of a row from A and a column from B.

Example Calculation

Let's multiply two 2×2 matrices:

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

AB = \begin{bmatrix} (1×5 + 2×7) & (1×6 + 2×8) \\ (3×5 + 4×7) & (3×6 + 4×8) \end{bmatrix}

AB = \begin{bmatrix} 19 & 22 \\ 43 & 50 \end{bmatrix}

Visual Representation

This diagram represents the multiplication of two matrices A and B.