9th Class Math Chapter 1 Solution covers Unit 1, Matrices and Determinants, one of the foundational chapters in the 9th class mathematics syllabus. This chapter introduces students to matrices, their types, operations, and how to solve simultaneous linear equations using matrix methods.
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These solutions explain every exercise from 1.1 to 1.6 in a clear, step-by-step format, making it easier for students to understand each concept and prepare confidently for exams.
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Introduction to Matrices
A matrix is a rectangular array of numbers arranged in rows and columns, enclosed in brackets. The idea of matrices was given by Arthur Cayley, an English mathematician who first developed the “Theory of Matrices” in 1858.
The 9th Class Math Chapter 1 Solution begins with defining a matrix and understanding its basic structure before moving into more advanced operations.
Order of a Matrix
The order of a matrix specifies the number of rows and columns it contains. If a matrix M has m rows and n columns, it is said to be of order m-by-n.
For example, a matrix with 2 rows and 3 columns has the order 2-by-3.
Equal Matrices
Two matrices A and B are equal only if:
- The order of A equals the order of B
- Their corresponding entries are equal
Exercise 1.1: Order and Equality of Matrices
This exercise from the 9th Class Math Chapter 1 Solution focuses on identifying the order of different matrices and checking whether given matrices are equal. Students practice finding unknown variables by comparing corresponding entries of equal matrices.
Types of Matrices Explained
Understanding different types of matrices is essential for solving problems in this chapter. The 9th Class Math Chapter 1 Solution covers the following types in detail:
Row and Column Matrix
- Row Matrix – has only one row (example: M = [2, -1, 7])
- Column Matrix – has only one column
Rectangular and Square Matrix
- Rectangular Matrix – the number of rows is not equal to the number of columns
- Square Matrix – the number of rows is equal to the number of columns
Null, Diagonal, and Identity Matrix
- Null (Zero) Matrix – every entry is 0
- Diagonal Matrix – at least one diagonal entry is non-zero, and all non-diagonal entries are zero
- Scalar Matrix – a diagonal matrix where all diagonal entries are the same and non-zero
- Identity Matrix – a diagonal matrix where all diagonal entries are 1, denoted by I
Symmetric and Skew-Symmetric Matrix
- Symmetric Matrix – a square matrix where A transpose equals A (Aᵗ = A)
- Skew-Symmetric Matrix – a square matrix where A transpose equals negative A (Aᵗ = -A)
Transpose and Negative of a Matrix
The transpose of a matrix is obtained by interchanging its rows into columns. The negative of a matrix is obtained by changing the signs of all its entries.
Exercise 1.2: Identifying Matrix Types
In this exercise, the 9th Class Math Chapter 1 Solution helps students identify unit matrices, row matrices, column matrices, null matrices, square matrices, rectangular matrices, diagonal matrices, and scalar matrices from a given set.
Addition, Subtraction, and Scalar Multiplication
Matrices must have the same order to be conformable for addition or subtraction. Addition is done by adding corresponding entries, while subtraction involves subtracting corresponding entries.
Laws of Matrix Addition
The 9th Class Math Chapter 1 Solution explains key laws verified through examples:
- Commutative Law – A + B = B + A
- Associative Law – (A + B) + C = A + (B + C)
- Additive Identity – A + O = A = O + A
- Additive Inverse – A + B = O = B + A
Scalar Multiplication of a Matrix
Multiplying a matrix by a real number k means multiplying every entry of the matrix by k. This is denoted as kA.
Exercise 1.3: Addition and Additive Inverse
This exercise covers checking which matrices are conformable for addition, finding additive inverses, and simplifying matrix expressions using addition and scalar multiplication rules.
Multiplication of Matrices
Two matrices A and B are conformable for multiplication only if the number of columns of A equals the number of rows of B. The 9th Class Math Chapter 1 Solution explains this rule with multiple solved examples.
Key Laws of Matrix Multiplication
- Associative Law – (AB)C = A(BC)
- Distributive Laws – A(B + C) = AB + AC and (A + B)C = AC + BC
- Commutative Law does NOT generally hold – AB ≠ BA in most cases
Multiplicative Identity
If AB = A = BA, then matrix B is called the identity matrix of A under multiplication.
Verification of (AB)ᵗ = Bᵗ Aᵗ
An important identity covered in the 9th Class Math Chapter 1 Solution shows that the transpose of a product equals the product of transposes in reverse order.
Exercise 1.4: Matrix Multiplication Practice
This exercise focuses on checking whether products of matrices are conformable for multiplication, finding products of given matrices, and verifying multiplication laws like associativity and distributivity.
Determinants, Singular and Non-Singular Matrices
The determinant of a 2-by-2 matrix A is calculated as ad – bc, denoted by |A| or det A.
Singular and Non-Singular Matrix
- Singular Matrix – a square matrix where the determinant equals zero (|A| = 0)
- Non-Singular Matrix – a square matrix where the determinant is not equal to zero (|A| ≠ 0)
Adjoint and Inverse of a Matrix
The adjoint of a matrix is obtained by interchanging the diagonal entries and changing the sign of the other entries. The 9th Class Math Chapter 1 Solution shows that:
M⁻¹ = Adj M / |M|
Inverse of a matrix is only possible if the matrix is non-singular.
Exercise 1.5: Determinants and Inverses
This exercise covers finding determinants of given matrices, checking whether matrices are singular or non-singular, and finding multiplicative inverses using the adjoint method.
Solving Simultaneous Linear Equations Using Matrices
One of the most practical applications in the 9th Class Math Chapter 1 Solution is solving systems of linear equations using two methods:
Matrix Inversion Method
For a system ax + by = m and cx + dy = n, written as AX = B, the solution is found using:
X = A⁻¹B
Cramer’s Rule
Using Cramer’s Rule, the solution is found as:
- x = |Aₓ| / |A|
- y = |Aᵧ| / |A|
where Aₓ and Aᵧ are formed by replacing the respective column with the constant matrix.
[internal link: 9th Class Math Chapter 4 Solution]
Exercise 1.6: Solving Word Problems Using Matrices
This final exercise in the 9th Class Math Chapter 1 Solution applies matrix methods to real-world problems, including:
- Finding dimensions of rectangles given perimeter and side relationships
- Finding triangle angles using algebraic conditions
- Solving speed and distance problems involving two moving objects
Students use both the matrix inversion method and Cramer’s rule to solve these practical applications, reinforcing their understanding of the entire chapter.
Suggested Image Alt Text: 9th Class Math Chapter 1 Solution – Matrices and Determinants Examples
FAQs
Q1: What does 9th Class Math Chapter 1 Solution cover?
These solutions cover Matrices and Determinants, including matrix types, addition, subtraction, multiplication, transpose, determinants, adjoint, inverse, and solving simultaneous linear equations using matrices.
Q2: What is the difference between a singular and non-singular matrix?
A singular matrix has a determinant equal to zero, meaning its inverse doesn’t exist. A non-singular matrix has a determinant not equal to zero, so its inverse can be calculated.
Q3: Does matrix multiplication follow the commutative law?
No, matrix multiplication generally does not follow the commutative law. In most cases, AB is not equal to BA, though it may hold true in specific special cases involving certain matrices.
Q4: What are the two methods for solving simultaneous linear equations using matrices?
The two methods are the Matrix Inversion Method, where X = A⁻¹B, and Cramer’s Rule, which uses determinants of modified matrices to find the values of x and y directly.
Q5: What is the formula for finding the determinant of a 2-by-2 matrix?
For a 2-by-2 matrix with entries a, b, c, and d, the determinant is calculated as ad – bc. This value determines whether the matrix is singular or non-singular.
Q6: How can 9th Class Math Chapter 1 Solution help in exam preparation?
These solutions provide step-by-step exercise answers with clear explanations, helping students understand matrix concepts thoroughly and solve similar problems confidently during board exams.