Zero Matrix (Null Matrix) | Matrices and Determinants | Short Introduction | Null Matrix Explained

Welcome to Everest Online Academy! In this video, we explore Zero Matrix (Null Matrix) , a fundamental concept in the Class 12 Business Mathematics syllabus under the chapter Matrices and Determinants . Understanding the properties and applications of the zero matrix is crucial for solving problems involving matrices, especially in business contexts like inventory management, cost analysis, and optimization. Whether you're struggling with definitions or need help applying the zero matrix in numerical problems, this video will guide you step-by-step with clear explanations and solved examples from past NEB papers .

✅ What You’ll Learn in This Video:

Introduction to Zero Matrix (Null Matrix) :
What is a Zero Matrix?
Definition: A matrix in which all elements are zero is called a Zero Matrix or Null Matrix .
Example:
O=[
0
0
​

0
0
​
]
Notation: Denoted by O or [0]
m×n
​
, where m is the number of rows and n is the number of columns.
Importance: Acts as the additive identity in matrix addition, similar to how "0" behaves in regular arithmetic.
Key Concepts :
Properties of Zero Matrix :
Additive Identity:
For any matrix A,
A+O=A
Multiplicative Annihilator:
When a matrix is multiplied by a zero matrix, the result is always a zero matrix:
A⋅O=OandO⋅A=O
Transpose: The transpose of a zero matrix is also a zero matrix:
O
T
=O
Applications of Zero Matrix :
Solving Systems of Linear Equations: Used as placeholders or initial values in iterative methods.
Business Applications: Represents situations where no data exists (e.g., zero inventory, zero production).
Matrix Operations: Simplifies calculations in addition, subtraction, and multiplication.
Solved Examples :
Example Problem 1 :
If A=[
1
3
​

2
4
​
] and O=[
0
0
​

0
0
​
], find A+O.
Solution:
A+O=[
1
3
​

2
4
​
]+[
0
0
​

0
0
​
]=[
1
3
​

2
4
​
]
Hence, A+O=A.
Example Problem 2 :
If B=[
5
7
​

6
8
​
] and O=[
0
0
​

0
0
​
], find B⋅O.
Solution:
B⋅O=[
5
7
​

6
8
​
]⋅[
0
0
​

0
0
​
]=[
0
0
​

0
0
​
]
Hence, B⋅O=O.
Old NEB Questions Solved :
Example Problem 1 (NEB Past Question):
Prove that the product of any matrix and a zero matrix is always a zero matrix.
Solution:
Step 1: Let A be any m×n matrix and O be a n×p zero matrix.
Step 2: Perform matrix multiplication:
(A⋅O)
ij
​
=
k=1
∑
n
​
A
ik
​
⋅O
kj
​

Since all elements of O are zero, each term in the summation becomes zero.
Step 3: Therefore, A⋅O=O. Similarly, O⋅A=O.
Hence, proved.
Example Problem 2 (NEB Past Question):
If X=[
0
0
​

0
0
​
], find X
T
.
Solution:
Step 1: Transpose the matrix:
X
T
=[
0
0
​

0
0
​
]
Step 2: Hence, X
T
=X.
Step-by-Step Solutions to NEB Past Year Questions :
Solving numerical problems involving zero matrices.
Application of concepts in real exam questions.
Common Mistakes to Avoid While Working with Zero Matrices
Tips to Score Full Marks in Business Mathematics Questions
Practice Exercises for Better Understanding
💡 Why Watch This Video?
The Zero Matrix (Null Matrix) is a frequently tested topic in the Class 12 Business Mathematics exam, especially in the context of matrix operations and their properties. By mastering this concept, you can confidently tackle exam questions and secure full marks in this section. This video simplifies the theory, provides solved examples, and includes practical tips to ensure you’re fully prepared.

📞 For more information, call us at 9767479277.
Thank you for choosing Everest Online Academy as your learning partner. Keep watching, keep learning, and keep achieving greatness! 🌟

#Class12BusinessMaths #ZeroMatrix #NullMatrix #NEBExamPreparation #EverestOnlineAcademy #StudyWithMe

zero matrix class 12, null matrix explained, business maths nepal, neb business mathematics questions, old question explained, everest online academy, business math tips and tricks, class 12 business math notes, matrices and determinants, matrix operations, additive identity, multiplicative annihilator, nepali students guide, neb preparation 2024, mastering business math, business math past papers, business math solutions, business math guide, business math strategies, business math practice, business math doubt clearing, business math discussion, business math interactive, business math learning

#Class12BusinessMaths #ZeroMatrix #NullMatrix #NEBExamPreparation #EverestOnlineAcademy #StudyWithMe #EducationInNepal #OnlineLearning #StudySmart #MathBasics #Class12SEE #MathMadeEasy #MathSuccess #LearnMath #MasterMath #ExamPrep2024 #MatrixOperations #AdditiveIdentity #MultiplicativeAnnihilator

Видео Zero Matrix (Null Matrix) | Matrices and Determinants | Short Introduction | Null Matrix Explained канала Everest Online Academy