RD Sharma Solutions for Class 12 Maths Chapter 5 - Algebra of Matrices - Free PDF Download
FAQs on RD Sharma Class 12 Maths Solutions Chapter 5 - Algebra of Matrices
1. What are the key topics covered in the RD Sharma Solutions for Class 12 Maths Chapter 5, Algebra of Matrices?
The RD Sharma Solutions for Class 12 Maths Chapter 5 provide comprehensive coverage of the Algebra of Matrices syllabus for the 2025-26 session. Key topics include:
Introduction to matrices, their order, and notation.
Different types of matrices, such as row, column, square, diagonal, scalar, identity, and zero matrices.
The concept of equality of matrices and how to solve for variables.
Operations on matrices, including addition, subtraction, scalar multiplication, and the multiplication of matrices.
Important properties of matrix operations, including the non-commutative nature of matrix multiplication.
The transpose of a matrix, along with its properties.
Concepts of symmetric and skew-symmetric matrices.
Finding the inverse of a matrix using elementary row and column operations.
2. What is the correct method for multiplying two matrices, A and B, as per the solutions?
The correct method for multiplying two matrices, A and B, to get the product AB, requires a specific condition and process. First, you must ensure that the number of columns in the first matrix (A) is exactly equal to the number of rows in the second matrix (B). If this condition is not met, the matrices cannot be multiplied. If it is met, each element of the resulting product matrix is calculated by taking the dot product of the corresponding row from matrix A and the column from matrix B. For instance, the element in the i-th row and j-th column of the product is found by multiplying each element of the i-th row of A with the corresponding element of the j-th column of B and then summing these products.
3. Why is matrix multiplication generally not commutative (i.e., AB ≠ BA)?
Matrix multiplication is not commutative for two primary reasons. Firstly, the existence of the product itself is directional. The product AB might be defined, but the product BA may not even be possible to calculate. This occurs if the number of columns in B does not match the number of rows in A. Secondly, even when both AB and BA are defined and result in matrices of the same order, the final matrices are usually different. This is because the calculation for an element in AB involves rows of A and columns of B, while the calculation for BA uses rows of B and columns of A, which is a fundamentally different operation that yields different results.
4. How can we use the concept of equality of matrices to solve for unknown variables?
To solve for unknown variables using the equality of matrices, two fundamental conditions must be met. First, the two matrices must be of the exact same order (i.e., have the same number of rows and columns). Second, if two matrices are equal, then their corresponding elements must also be equal. The step-by-step process is:
Confirm both matrices have identical dimensions (e.g., both are 2x3).
Create a system of equations by setting the corresponding elements from both matrices equal to each other.
Solve these equations to find the values of the unknown variables (e.g., x, y, z).
5. What is the key difference between a symmetric and a skew-symmetric matrix?
The key difference between these two types of square matrices lies in their relationship with their transpose. A square matrix 'A' is defined as symmetric if it is identical to its transpose (A = A'). In a symmetric matrix, the element in the i-th row and j-th column (aij) is equal to the element in the j-th row and i-th column (aji). Conversely, a square matrix 'A' is skew-symmetric if it is equal to the negative of its transpose (A = -A'). This means aij = -aji for all i and j. A critical consequence of this property is that all diagonal elements of a skew-symmetric matrix must be zero.
6. Is it possible for the product of two non-zero matrices to be a zero matrix?
Yes, it is entirely possible for the product of two non-zero matrices to result in a zero matrix. This is a unique and important property of matrix multiplication that distinguishes it from the multiplication of real numbers, where if ab=0, either a or b must be zero. In matrix algebra, you can have two matrices, A and B, where neither is a zero matrix, yet their product AB equals a zero matrix (a matrix where all elements are zero). This demonstrates that the standard cancellation law does not apply to matrices.
7. How are elementary row operations used to find the inverse of a matrix?
Elementary row operations are used to find the inverse of a square matrix 'A' by systematically transforming it into the identity matrix 'I'. The process begins by creating an augmented matrix of the form [A | I]. A sequence of elementary row operations—such as swapping two rows, multiplying a row by a non-zero constant, or adding a multiple of one row to another—is applied to the left side (matrix A) with the goal of converting it into 'I'. Crucially, every operation performed on the left side must also be performed on the right side (matrix I). Once the left side becomes the identity matrix, the right side will have been transformed into the inverse of A (A⁻¹). The final augmented matrix will be in the form [I | A⁻¹].
8. Why should a Class 12 student use RD Sharma solutions in addition to the NCERT textbook for Algebra of Matrices?
While the NCERT textbook builds a strong conceptual foundation, the RD Sharma solutions for Algebra of Matrices offer two main advantages for thorough exam preparation. Firstly, RD Sharma provides a significantly larger volume and variety of problems, from basic to advanced levels, which helps in mastering concepts and improving problem-solving speed. Secondly, it often includes a wider range of question typologies, including Higher Order Thinking Skills (HOTS) problems, which better prepare students for the full spectrum of questions they might encounter in the CBSE Class 12 board exams and other competitive entrance tests.
