Matrix Addition and Subtraction

In linear algebra, matrix addition and subtraction are fundamental operations that involve adding or subtracting corresponding elements of matrices. These operations are only defined for matrices of the same size.

Matrix Addition

Given two $m\times n$ matrices $A=[a_{ij}]$ and $B=[b_{ij}]$, the sum $A+B$ is defined as the matrix $C=[c_{ij}]$ where $c_{ij}=a_{ij}+b_{ij}$ for all $1\le i\le m$, $1\le j\le n$.

Properties of Matrix Addition

1. Commutative Property: $A+B=B+A$ for any matrices $A$ and $B$ where $A$ and $B$ are of the same size.
2. Associative Property: $(A+B)+C=A+(B+C)$ for any matrices $A$, $B$, and $C$ of the same size.
3. Identity Element: The zero matrix, denoted by $0$, acts as the additive identity matrix in matrix addition. For any matrix $A$, $A+0=A$.

Matrix Subtraction

Matrix subtraction is defined similarly to matrix addition. Given two $m\times n$ matrices $A=[a_{ij}]$ and $B=[b_{ij}]$, the difference $A-B$ is defined as the matrix $C=[c_{ij}]$ where $c_{ij}=a_{ij}-b_{ij}$ for all $1\le i\le m$, $1\le j\le n$.

Properties of Matrix Subtraction

1. Non-commutativity: Matrix subtraction is not commutative, i.e., $A-B\ne B-A$ in general.
2. Inverse Element: For any matrix $A$, there exists a matrix $B$ such that $A+B=0$, where $0$ is the zero matrix. The matrix $B$ is called the additive inverse of $A$ and is denoted by $-A$.

Example

Let’s consider the following matrices:

A=\begin{bmatrix} 1& 2\\ 3& 4\end{bmatrix}\quad\text{and}\quad B=\begin{bmatrix} 4& 3\\ 2& 1\end{bmatrix}$$ The sum $A+ B$ is:

A+ B=\begin{bmatrix} 1+ 4& 2+ 3\ 3+ 2& 4+ 1\end{bmatrix}=\begin{bmatrix} 5& 5\ 5& 5\end{bmatrix}$$

The difference $A-B$ is:

A- B=\begin{bmatrix} 1- 4& 2- 3\\ 3- 2& 4- 1\end{bmatrix}=\begin{bmatrix}-3&-1\\ 1& 3\end{bmatrix}$$ ## Historical Context Matrix operations, including addition and subtraction, were first introduced by the mathematician James Joseph Sylvester in the mid-19th century. Matrices have since become a fundamental tool in various fields such as physics, computer science, economics, and engineering. ## Real-life Example Consider a scenario where a company tracks its monthly sales data using matrices. The matrix $A$ represents the sales data for the first quarter of the year, and the matrix $B$ represents the sales data for the second quarter. By adding the matrices $A$ and $B$, the company can obtain the total sales for the first half of the year. Similarly, by subtracting the matrices $A$ and $B$, the company can analyze the difference in sales between the two quarters. ## Exam Questions 1. Given matrices $A=\begin{bmatrix} 2& 3\\ 4& 1\end{bmatrix}$ and $B=\begin{bmatrix} 1& 1\\ 2& 2\end{bmatrix}$, calculate $A+ B$. 2. Show that matrix addition is commutative using arbitrary matrices $A$ and $B$ of the same size. 3. Define the zero matrix and explain its role in matrix addition.