Maths Class 12 Chapter 3 Questions and Answers - Free PDF Download
In NCERT Solutions for Class 12 Maths Chapter 3 Matrices Ex 3.4, you’ll dive into important concepts like invertible matrices and finding the inverse of a matrix. This part of the chapter helps you learn tricks for solving linear equations and understanding matrix properties, which are super useful for board exams and future studies. If you feel confused by terms like ‘identity matrix’ or ‘inverse,’ don’t worry—Vedantu makes each step simple and clear.
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With these NCERT Solutions, practicing becomes easier. You can check the detailed solutions anytime and download the free PDFs to study offline. Plus, if you want a quick look at your Maths syllabus, check out the CBSE Class 12 Maths Syllabus.
This chapter carries 5 marks in your CBSE exam, so learning these concepts well can really boost your score. Want more help? Try the full set of NCERT Maths Solutions for Class 12 to strengthen your basics and exam confidence.
FAQs on NCERT Solutions For Class 12 Maths Chapter 3 Matrices Exercise 3.4 - 2025-26
1. What are the fundamental operations on matrices covered in NCERT Class 12 Chapter 3?
As per the CBSE 2025-26 syllabus, the NCERT solutions for Class 12 Maths Chapter 3 detail three fundamental algebraic operations on matrices:
- Addition of matrices: This involves adding the corresponding elements of two matrices that must be of the same order.
- Multiplication by a scalar: This operation involves multiplying every element of the matrix by a constant number (a scalar).
- Multiplication of matrices: This involves a row-by-column multiplication. For the product AB to be defined, the number of columns in matrix A must equal the number of rows in matrix B.
2. What is the correct step-by-step method for multiplying two matrices, A and B?
To correctly multiply two matrices, A and B, as per the NCERT methodology, you must first ensure they are compatible for multiplication: the number of columns in matrix A must equal the number of rows in matrix B. If this condition isn't met, the product is not defined.
The steps are as follows:
- To find the element in the i-th row and j-th column of the product matrix, select the i-th row of matrix A and the j-th column of matrix B.
- Multiply their corresponding elements one by one.
- Sum up all these products. This final sum gives you the single element for that specific position in the resulting matrix.
3. How do you find the inverse of a square matrix using elementary row operations as per the NCERT methodology?
To find the inverse of a square matrix A using elementary row operations, start by writing the equation A = IA, where I is the identity matrix of the same order as A.
The step-by-step method is as follows:
- Apply a sequence of elementary row operations to the matrix A on the left-hand side (LHS) of the equation to transform it into the identity matrix I.
- Simultaneously, apply the exact same sequence of operations to the identity matrix I on the right-hand side (RHS).
- Once the matrix A on the LHS becomes the identity matrix I, the matrix on the RHS will be transformed into the inverse of A, or A⁻¹.
The final equation will be I = A⁻¹A. If you obtain a row or column of all zeros on the LHS at any stage, then A⁻¹ does not exist.
4. What are invertible matrices, and how can you determine if a matrix is invertible?
An invertible matrix is a square matrix 'A' for which another square matrix 'B' of the same order exists, such that their product is the identity matrix (I). This is expressed as AB = BA = I. The matrix B is called the inverse of A, written as A⁻¹.
According to the NCERT syllabus, you can determine if a matrix is invertible in two main ways:
- Using elementary operations: A square matrix A is invertible if it can be reduced to the identity matrix (I) by applying a sequence of elementary operations.
- Using determinants (Chapter 4): A square matrix A is invertible if and only if its determinant is non-zero (det(A) ≠ 0).
5. What are the three elementary row operations used to solve problems in Chapter 3?
The NCERT solutions for Chapter 3 are based on three specified elementary row operations (or transformations) used to find a matrix's inverse. They are:
- Interchange of any two rows: Swapping the positions of any two rows (denoted as Rᵢ ↔ Rⱼ).
- Multiplication of a row by a non-zero scalar: Multiplying all elements of a single row by any non-zero constant 'k' (denoted as Rᵢ → kRᵢ).
- Addition of a multiple of one row to another: Adding the elements of one row, multiplied by a non-zero scalar 'k', to the corresponding elements of another row (denoted as Rᵢ → Rᵢ + kRⱼ).
These same three operations can also be applied to columns.
6. Why is matrix multiplication not always commutative (i.e., AB ≠ BA)?
Matrix multiplication is not always commutative due to its row-by-column definition. Unlike the multiplication of real numbers, the order in which matrices are multiplied significantly changes the result. There are two primary reasons:
- Existence of the Product: The product AB might be defined while BA is not. For instance, if A is a 2x3 matrix and B is a 3x4 matrix, AB is a valid 2x4 matrix. However, BA is undefined as the number of columns in B (4) does not match the number of rows in A (2).
- Different Resulting Matrices: Even when both AB and BA are defined (e.g., for two square matrices of the same order), the resulting matrices are generally not equal because the calculation process yields different elements when the order is reversed. This property is known as non-commutativity.
7. Why can't a rectangular matrix have an inverse?
A rectangular matrix cannot have an inverse because the definition of an inverse requires that for a matrix A, there exists a matrix B such that AB = BA = I, where I is the identity matrix. This condition can only be satisfied if both A and B are square matrices of the same order.
If A is a rectangular matrix of order m x n (where m ≠ n), any potential inverse B would have to be of order n x m. The product AB would be an m x m matrix, while the product BA would be an n x n matrix. Since m ≠ n, the resulting matrices AB and BA have different orders and can never be equal, making an inverse impossible.
8. How is the concept of an inverse matrix (A⁻¹) used to solve a system of linear equations?
The inverse matrix provides a direct method for solving a system of linear equations. A system can be represented in matrix form as AX = B, where A is the coefficient matrix, X is the variable matrix, and B is the constant matrix.
If the coefficient matrix A is invertible and its inverse A⁻¹ exists, you can find the solution by pre-multiplying both sides of the equation by A⁻¹:
A⁻¹(AX) = A⁻¹B
Since A⁻¹A = I (the identity matrix), the equation simplifies to IX = A⁻¹B, which gives the unique solution for the variables as X = A⁻¹B.
9. What is the significance of the transpose of a matrix (A'), and how is it used to define symmetric and skew-symmetric matrices?
The transpose of a matrix (A' or Aᵀ) is an operation where the matrix's rows and columns are interchanged. Its significance lies in its use for defining special types of matrices as per the NCERT syllabus.
The transpose helps define:
- Symmetric Matrix: A square matrix A is symmetric if it is equal to its own transpose (A' = A). This means the element at position (i, j) is equal to the element at (j, i).
- Skew-Symmetric Matrix: A square matrix A is skew-symmetric if it is the negative of its transpose (A' = -A). This means the element at (i, j) is the negative of the element at (j, i), and all diagonal elements must be zero.
A key theorem in Chapter 3 also states that any square matrix can be uniquely expressed as the sum of a symmetric and a skew-symmetric matrix.
