Introduction to Adjoint and Inverse of a Matrix - Free PDF Download
FAQs on RD Sharma Class 12 Maths Solutions Chapter 7 - Adjoint and Inverse of a Matrix
1. What are the key concepts covered in RD Sharma Class 12 Maths Solutions for Chapter 7, Adjoint and Inverse of a Matrix?
These solutions primarily focus on the core concepts required for the CBSE 2025-26 syllabus. Key topics include:
- Defining the adjoint of a square matrix and its properties.
- Calculating the inverse of a square matrix.
- Understanding the conditions for a matrix to be invertible (i.e., being non-singular).
- Using the inverse of a matrix to solve systems of linear equations.
- Applying properties of adjoints and inverses to solve complex problems.
2. What is the adjoint of a square matrix and how is it calculated?
The adjoint of a square matrix A, denoted as adj(A), is the transpose of its cofactor matrix. To calculate it, you follow these steps:
- First, find the minor for each element of the matrix A.
- Next, determine the cofactor for each element by applying the sign rule: Cij = (-1)i+j Mij, where Mij is the minor of the element aij.
- Arrange these cofactors into a new matrix, called the cofactor matrix.
- Finally, find the transpose of this cofactor matrix to get the adjoint of A.
3. What is the formula to find the inverse of a matrix using its adjoint?
The inverse of a square matrix A, denoted as A-1, can be found using the formula: A-1 = (1/|A|) × adj(A). Here, |A| represents the determinant of the matrix A, and adj(A) is the adjoint of matrix A. An inverse only exists if the determinant |A| is not equal to zero.
4. What are the detailed steps to calculate the inverse of a 3x3 matrix?
To find the inverse of a 3x3 matrix, follow this systematic procedure:
- Step 1: Calculate the determinant of the matrix. If the determinant is zero, the matrix is singular, and its inverse does not exist.
- Step 2: Find the matrix of cofactors. For each element in the original matrix, calculate its corresponding cofactor.
- Step 3: Determine the adjoint of the matrix by taking the transpose of the cofactor matrix.
- Step 4: Finally, calculate the inverse by dividing the adjoint matrix by the determinant: A-1 = (1/|A|) × adj(A).
5. How is the inverse of a matrix used to solve a system of linear equations?
A system of linear equations like a₁x + b₁y + c₁z = d₁, a₂x + b₂y + c₂z = d₂, and a₃x + b₃y + c₃z = d₃ can be written in matrix form as AX = B. Here, A is the coefficient matrix, X is the variable matrix, and B is the constant matrix. To find the solution, you can pre-multiply by the inverse of A: X = A-1B. This method provides a unique solution, provided that A is a non-singular matrix.
6. Why must a matrix be non-singular to have an inverse?
A matrix must be non-singular (i.e., its determinant must be non-zero) for its inverse to exist. This is because the formula for the inverse is A-1 = (1/|A|) × adj(A). If the determinant |A| were zero, the term 1/|A| would be undefined, making it impossible to calculate the inverse. A non-singular matrix ensures that a unique solution exists for the corresponding system of linear equations.
7. What is the difference between the transpose and the adjoint of a matrix?
The transpose and adjoint are both derived from a square matrix but are fundamentally different. The transpose, denoted as AT, is found by simply interchanging the rows and columns of the matrix A. The adjoint, adj(A), is a more complex calculation; it is the transpose of the matrix of cofactors of A. While the transpose is a simple rearrangement, the adjoint requires calculating minors and cofactors first.
8. How do RD Sharma Solutions for this chapter help in scoring more marks in exams?
RD Sharma Solutions for Chapter 7 are beneficial for exam preparation as they offer a vast number of practice questions, which helps in mastering the concepts of adjoint and inverse. The solutions provide step-by-step methods for complex problems, aligning with the CBSE marking scheme. By practising these questions, students can improve their problem-solving speed and accuracy, build confidence in handling different types of questions, and gain a deeper understanding of how to apply formulas correctly in the board exam.
9. Is it possible to find the inverse of a matrix if it satisfies a given matrix equation?
Yes, if a square matrix A satisfies a characteristic polynomial equation (e.g., A² - 5A + 7I = 0), you can find its inverse without calculating the adjoint. To do this, pre-multiply or post-multiply the entire equation by A-1. For the example A² - 5A + 7I = 0, multiplying by A-1 gives A - 5I + 7A-1 = 0. You can then isolate A-1 to get A-1 = (1/7)(5I - A). This method is often quicker for specific exam questions.
