RD Sharma Class 12 Solutions Chapter 8 - Solution of Simultaneous Linear Equations (Ex 8.1) Exercise 8.1 - Free PDF Download
FAQs on RD Sharma Class 12 Solutions Chapter 8 - Solution of Simultaneous Linear Equations (Ex 8.1) Exercise 8.1
1. What is the correct method to solve a system of three linear equations using matrices as given in RD Sharma Class 12 Solutions for Exercise 8.1?
To solve a system of three linear equations like a₁x + b₁y + c₁z = d₁, a₂x + b₂y + c₂z = d₂, and a₃x + b₃y + c₃z = d₃, you should follow the matrix inverse method, which is represented as AX = B.
First, express the equations in matrix form, where A is the coefficient matrix, X is the variable matrix, and B is the constant matrix.
Next, calculate the determinant of A, which is denoted as |A|.
If |A| ≠ 0, find the inverse of A using the formula A⁻¹ = (1/|A|) * adj(A).
Finally, the solution is given by the equation X = A⁻¹B. The RD Sharma solutions provide a clear, step-by-step application of this entire method.
2. How do I check if a system of linear equations is consistent or inconsistent as per the methods in RD Sharma Chapter 8?
To check for consistency, you must first calculate the determinant of the coefficient matrix A, denoted as |A|. The conditions are as follows:
If |A| ≠ 0, the system is consistent and has a unique solution.
If |A| = 0, you must then calculate the product (adj A)B.
If (adj A)B = O (where O is a null matrix), the system is consistent and has infinitely many solutions.
If (adj A)B ≠ O, the system is inconsistent and has no solution. These checks are fundamental to solving problems in Exercise 8.1.
3. What is the significance of finding the adjoint of a matrix (adj A) when solving simultaneous linear equations?
The adjoint of a matrix, adj(A), is crucial because it is required to find the inverse of the matrix A. The formula for a matrix inverse is A⁻¹ = (1/|A|) * adj(A). Since the unique solution to a system of equations AX = B is given by X = A⁻¹B, you cannot find the solution using this method without first calculating the adjoint. It serves as an essential intermediate step in the matrix inverse method taught in this chapter.
4. In the RD Sharma solutions for Exercise 8.1, what does it mean if the determinant of the coefficient matrix |A| is zero?
If the determinant of the coefficient matrix |A| is zero, it signifies that the system of equations does not have a unique solution. This leads to two distinct possibilities:
The system may have infinitely many solutions (a consistent but dependent system).
The system may have no solution (an inconsistent system).
To determine which case applies, you must proceed to the next step of calculating (adj A)B. A zero determinant indicates that the matrix A is singular, and therefore its inverse does not exist.
5. Why is the matrix method often preferred over Cramer's Rule for solving a system of three linear equations?
While both methods are valid, the matrix method (AX = B) is often considered more systematic and less prone to calculation errors for 3x3 systems. Cramer's Rule requires the calculation of four separate 3x3 determinants (for D, Dₓ, Dᵧ, and D₂), which can be tedious and increases the chances of error. In contrast, the matrix method involves finding only one determinant and one adjoint, followed by a single matrix multiplication, providing a more organised workflow for complex problems.
6. What are the first steps to take when starting a problem from RD Sharma Class 12 Maths Exercise 8.1?
The first and most important step is to accurately write the given system of linear equations in the standard matrix form, AX = B.
Matrix A: This is the matrix formed by the coefficients of the variables x, y, and z.
Matrix X: This is the column matrix of the variables themselves (e.g., [x, y, z]ᵀ).
Matrix B: This is the column matrix of the constant terms from the right-hand side of the equations.
Correctly setting up these three matrices is essential before proceeding with any calculations.
7. How do the RD Sharma Solutions for Class 12 Chapter 8 help in mastering questions for the CBSE board exam?
The RD Sharma Solutions for this chapter provide detailed, step-by-step procedures for solving systems of linear equations, a topic that frequently appears in the CBSE Class 12 board exams. By following these solutions, you learn the correct format and methodology expected by examiners, how to handle special cases like inconsistent systems, and how to minimise calculation errors. This practice ensures you can solve related board questions accurately and efficiently for maximum marks.
8. What is the difference between a homogeneous and a non-homogeneous system of linear equations?
The key difference between these systems lies in the constant terms, represented by the matrix B in the equation AX = B.
A non-homogeneous system has a non-zero constant matrix B (at least one constant is not zero). It can have a unique solution, no solution, or infinite solutions.
A homogeneous system has a constant matrix B that is a null matrix (all constants are zero). This type of system is always consistent and will either have a trivial solution (x=0, y=0, z=0) or infinitely many non-trivial solutions.
Exercise 8.1 primarily deals with solving non-homogeneous systems.
