Class 12 RS Aggarwal Chapter-7 Adjoint and Inverse of a Matrix Solutions - Free PDF Download
FAQs on RS Aggarwal Class 12 Solutions Chapter-7 Adjoint and Inverse of a Matrix
1. Where can I find reliable, step-by-step solutions for all questions in RS Aggarwal Class 12 Maths Chapter 7?
Vedantu provides comprehensive, step-by-step solutions for every question in RS Aggarwal Class 12 Maths Chapter 7, Adjoint and Inverse of a Matrix. These solutions are prepared by subject matter experts and are fully aligned with the CBSE 2025-26 syllabus, ensuring you understand the correct methodology for solving problems related to finding the adjoint, calculating the inverse, and solving systems of linear equations.
2. What is the correct step-by-step method to find the inverse of a 3x3 matrix using the adjoint method?
To find the inverse of a 3x3 matrix 'A' using the adjoint method, you must follow these precise steps as demonstrated in the RS Aggarwal solutions:
- Step 1: Calculate the determinant of the matrix, |A|. If |A| = 0, the matrix is singular and its inverse does not exist.
- Step 2: Find the cofactor for each element of the matrix A.
- Step 3: Construct the cofactor matrix by arranging the calculated cofactors in their corresponding positions.
- Step 4: Determine the adjoint of the matrix, adj(A), by taking the transpose of the cofactor matrix.
- Step 5: Use the formula A⁻¹ = (1/|A|) × adj(A) to calculate the inverse.
3. How are the key theorems on adjoint and inverse applied in the RS Aggarwal solutions for Chapter 7?
The solutions for RS Aggarwal Chapter 7 demonstrate the practical application of key theorems for verification and problem-solving. For instance, the theorem A(adj A) = (adj A)A = |A|I is frequently used to verify the correctness of a calculated adjoint. Similarly, the property that a matrix A is invertible if and only if it is non-singular (i.e., |A| ≠ 0) is the foundational check before attempting to find the inverse of any matrix in the exercises.
4. How is the inverse of a matrix used to solve a system of linear equations?
The inverse of a matrix is a powerful tool for solving a system of linear equations. The process, as detailed in Chapter 7 solutions, involves:
- Representing the system of equations in the matrix form AX = B, where A is the coefficient matrix, X is the variable matrix, and B is the constant matrix.
- Checking if the coefficient matrix A is non-singular by ensuring its determinant is not zero.
- Calculating the inverse of the coefficient matrix, A⁻¹.
- Finding the solution by pre-multiplying both sides by the inverse, which gives the formula X = A⁻¹B.
5. What is a common mistake students make when calculating the adjoint of a matrix and how can I avoid it?
A very common mistake is confusing the cofactor matrix with the adjoint matrix. Students often correctly calculate the matrix of cofactors but forget the crucial final step of taking its transpose to find the adjoint. To avoid this, always remember: adj(A) = (Cofactor Matrix)ᵀ. A good practice is to explicitly write down the cofactor matrix first and then carefully write down its transpose as the adjoint before using it in the inverse formula.
6. Why must a matrix be non-singular for its inverse to exist?
A matrix must be non-singular (meaning its determinant is not zero) for its inverse to exist because the formula for the inverse is A⁻¹ = (1/|A|) × adj(A). If the determinant |A| were zero, the term 1/|A| would become 1/0, which is undefined. This mathematical constraint makes it impossible to calculate the inverse. Conceptually, a singular matrix represents a system of linear equations that does not have a unique solution, which is why no inverse transformation exists.
7. What is the fundamental relationship between the adjoint of a matrix and its inverse?
The adjoint of a matrix and its inverse are intrinsically linked. The adjoint is a necessary intermediate step in the calculation of the inverse. The inverse is defined as the adjoint of the matrix scaled by the reciprocal of its determinant. In simple terms, the adjoint provides the structure of the inverse matrix, while the determinant provides the correct scaling factor. You cannot find the inverse using this method without first correctly computing the adjoint.
