Exercise 4.1 | New First Year Math 2025 | Sir Khawar Solution| New 11th Class Math 2025
This solution of Exercise 4.1 is prepared by Sir Khawar for the new syllabus of First Year Mathematics 2025 (11th class). The main topic of this exercise is Matrices and Its Types. We also study the basic operations of Addition and Subtraction of matrices. In this description, we will explain all concepts in simple words and with easy math symbols so that every student can understand without any difficulty.
Introduction
A matrix is a rectangular arrangement of numbers, symbols, or expressions written in rows and columns.
For example:
A =
This is a 2 × 3 matrix because it has 2 rows and 3 columns.
In general, a matrix is written as:
A =
where
i = row number
j = column number
aᵢⱼ = element in the i-th row and j-th column
Order of a Matrix
The order of a matrix is written as m × n, where
m = number of rows
n = number of columns
Example:
If a matrix has 3 rows and 4 columns, its order is 3 × 4.
Types of Matrices
In Exercise 4.1, we study different types of matrices. Let’s explain them one by one with examples.
1. Row Matrix
A matrix with only one row is called a row matrix.
Example:
A =
(Order: 1 × 3)
2. Column Matrix
A matrix with only one column is called a column matrix.
Example:
B =
(Order: 3 × 1)
3. Square Matrix
If the number of rows and number of columns are equal, then the matrix is called a square matrix.
Example:
C =
(Order: 2 × 2, rows = columns)
4. Zero Matrix (Null Matrix)
A matrix in which all elements are zero is called a zero matrix.
Example:
O =
5. Diagonal Matrix
If all non-diagonal elements are zero and diagonal elements can be any number, then the matrix is called a diagonal matrix.
Example:
D =
6. Scalar Matrix
A diagonal matrix in which all diagonal elements are equal is called a scalar matrix.
Example:
S =
7. Identity Matrix (Unit Matrix)
A square matrix in which all diagonal elements are 1 and all non-diagonal elements are 0 is called an identity matrix.
Example:
I =
8. Upper Triangular Matrix
A square matrix in which all elements below the diagonal are zero is called an upper triangular matrix.
Example:
U =
9. Lower Triangular Matrix
A square matrix in which all elements above the diagonal are zero is called a lower triangular matrix.
Example:
L =
10. Equal Matrices
Two matrices A and B are equal if:
1. Their orders are the same
2. Corresponding elements are equal
Example:
A = , B =
Here A = B
Operations on Matrices
In this exercise, we also study addition and subtraction of matrices.
1. Addition of Matrices
If A and B are two matrices of the same order, then their sum is obtained by adding corresponding elements.
Example:
A =
B =
A + B =
A + B =
2. Subtraction of Matrices
If A and B are two matrices of the same order, then their difference is obtained by subtracting corresponding elements.
Example:
A =
B =
A − B =
A − B =
Properties of Matrix Addition
1. Commutative Law:
A + B = B + A
2. Associative Law:
(A + B) + C = A + (B + C)
3. Existence of Zero Matrix:
A + O = A
4. Existence of Negative Matrix:
A + (−A) = O
Properties of Matrix Subtraction
1. A − B ≠ B − A (subtraction is not commutative)
2. (A − B) − C = A − (B + C)
Importance of Matrices
Matrices are very important in modern mathematics and have applications in:
Physics (mechanics, quantum physics)
Economics (input-output models)
Computer Science (graphics, algorithms)
Engineering (electrical networks, civil structures)
Statistics (data arrangement)
Understanding the basic operations in Exercise 4.1 helps students to solve bigger problems in later chapters.
PDF SOLUTION:
Step-by-Step Learning Strategy
1. First, understand the types of matrices with small examples.
2. Practice addition and subtraction using simple 2 × 2 and 3 × 3 matrices.
3. Remember that operations are only possible when the order of matrices is the same.
4. Solve all examples given in the textbook before attempting exercise questions.
The solutions are explained in an easy method so that even an average student can understand. Each question is solved step by step, keeping the definitions and rules in mind.
Students are advised to:
Read the definition of each type carefully
Write examples on their own
Practice at least 5 sums of addition and subtraction daily
Conclusion
Exercise 4.1 of First Year Math (New 2025 syllabus) introduces students to the basic concepts of matrices. This includes types of matrices, their properties, and the basic operations of addition and subtraction. With the help of Sir Khawar’s solution, students can easily learn these topics step by step.
This chapter forms the foundation for advanced topics in matrices like multiplication, determinants, inverse matrices, and applications which are discussed in later exercises. A strong grip on this exercise will make future learning easy and enjoyable.
#FirstYearMath2025 #11thClassMath #Exercise4_1 #Matrices #TypesOfMatrices #MatrixAddition #MatrixSubtraction #MathBySirKhawar #NewSyllabus2025 #MathUniverseOnline
0 Comments