Introduction

In linear algebra, matrices are rectangular arrays of numbers, and operations like addition, subtraction, and multiplication follow specific rules. Below, I’ll explain each operation clearly and concisely, assuming you’re familiar with basic matrix notation.

Matrix Addition

• Definition: Two matrices can be added if they have the same dimensions (same number of rows and columns). The result is a new matrix where each element is the sum of the corresponding elements in the input matrices.

Commutative Property:

 

$$A+B=B+A$$

 

2. Associative Property:

 

$$(A+B)+C=A+(B+C)$$

 

3. Additive Identity:
   There exists a zero matrix 0 such that:

 

$$A+0=A$$

 

4. Additive Inverse:
   For every matrix A , there exists a matrix −A such that:

$$A + (-A) = 0$$

 

 

 

 

Example of matrix Addition 

1.Example

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

\[
A + B = \begin{bmatrix}
1+5 & 2+6 \\
3+7 & 4+8
\end{bmatrix}
= \begin{bmatrix}
6 & 8 \\
10 & 12
\end{bmatrix}
\]

Matrix Subtraction

Matrix subtraction involves subtracting corresponding elements of two matrices. Like addition, both matrices must have the same dimensions.

Properties of Matrix Subtraction:

1. Not Commutative:

 

$$A-B\mathrm{\ne }B-A$$

 

2. Distributive over Scalar Multiplication:

 

$$k(A-B)=kA-kB$$

 

3. Identity Element:

 

$$A – 0 = A$$

 

Example of Matrix Subtraction

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

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

Matrix Multiplication

Matrix multiplication involves computing the dot product of the rows of the first matrix with the columns of the second. It is only defined if the number of columns in the first matrix equals the number of rows in the second matrix.

Properties of Matrix Multiplication:

1. Associative Property:

$$(A \cdot B) \cdot C = A \cdot (B \cdot C)$$

 

2. Distributive Over Addition:

$$A(B + C) = AB + AC$$

 

3. Not Commutative:

$$AB \neq BA \quad \text{(in general)}$$

 

4. Multiplicative Identity:
   For any matrix

   $A$,

$$AI=IA=A$$

 

where

$I$

is the identity matrix.

Example of Matrix Multiplication

\[
A = \begin{bmatrix} 2 & 3 \\ 1 & 4 \end{bmatrix}, \quad
B = \begin{bmatrix} 5 & 2 \\ 1 & 3 \end{bmatrix}
\]

\[
AB = \begin{bmatrix}
2 \cdot 5 + 3 \cdot 1 & 2 \cdot 2 + 3 \cdot 3 \\
1 \cdot 5 + 4 \cdot 1 & 1 \cdot 2 + 4 \cdot 3
\end{bmatrix}
= \begin{bmatrix}
10 + 3 & 4 + 9 \\
5 + 4 & 2 + 12
\end{bmatrix}
= \begin{bmatrix}
13 & 13 \\
9 & 14
\end{bmatrix}
\]

Applications:

These operations underpin many fields, including solving linear systems, computer graphics, data analysis, and machine learning. For example, matrix addition combines datasets, subtraction computes differences, and multiplication represents transformations or compositions.

• Non-Commutativity: Matrix multiplication’s non-commutative nature is a key distinction from scalar arithmetic, requiring careful attention to order in calculations.
• Computational Considerations: Matrix operations can be computationally intensive for large matrices, leading to specialized algorithms (e.g., Strassen’s algorithm for multiplication) in numerical linear algebra.