Addition of Matrices: A Fundamental Operation

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Matrix addition is one of the basic operations in linear algebra. It is straightforward and essential for various applications in mathematics, physics, engineering, and computer science. This guide will walk you through the concept and process of matrix addition.

Definition

Matrix addition involves adding two matrices by adding their corresponding elements. This operation is defined only for matrices of the same dimensions. If $A$ and $B$ are both $m\times n$ matrices, their sum $C$ is also an $m\times n$ matrix.

Process

To add two matrices $A$ and $B$, simply add each element in $A$ to the corresponding element in $B$. Mathematically, this is represented as:

Cij=Aij+BijC_{ij} = A_{ij} + B_{ij}Cij​=Aij​+Bij​

Example

Consider the following matrices $A$ and $B$:

A=(123456),B=(789101112)A = begin{pmatrix} 1 & 2 & 3 4 & 5 & 6 end{pmatrix}, quad B = begin{pmatrix} 7 & 8 & 9 10 & 11 & 12 end{pmatrix}A=(14​25​36​),B=(710​811​912​)

To find the sum $C=A+B$:

C=(1+72+83+94+105+116+12)C = begin{pmatrix} 1+7 & 2+8 & 3+9 4+10 & 5+11 & 6+12 end{pmatrix}C=(1+74+10​2+85+11​3+96+12​)

Calculating the elements:

C=(81012141618)C = begin{pmatrix} 8 & 10 & 12 14 & 16 & 18 end{pmatrix}C=(814​1016​1218​)

Properties

1. Commutativity: $A+B=B+A$
2. Associativity: $(A+B)+C=A+(B+C)$
3. Additive Identity: There exists an additive identity matrix $0$ such that $A+0=A$
4. Additive Inverse: For every matrix $A$, there exists a matrix $-A$ such that $A+(-A)=0$

Applications

• Linear Systems: Adding matrices is used to combine systems of linear equations.
• Data Analysis: In statistics and data analysis, matrix addition helps in combining data sets.
• Image Processing: Matrices representing images can be added to perform operations like blending.

Conclusion

Matrix addition is a simple yet powerful operation in linear algebra. Understanding how to add matrices and the properties of matrix addition is crucial for solving more complex mathematical problems and for applications in various scientific fields.