Matrices Notes: Class 12 Mathematics Chapter 3
Class 12 Mathematics Chapter 3 Matrices covers the definitions and types of matrices, operations such as addition and multiplication, symmetric and skew-symmetric matrices, and elementary operations used to find the inverse of a matrix.
Overview of Topics & Subtopics (Click to Jump)
1. Types and Operations on Matrices
Short Note:
• Matrices are classified by their order and structure into row, column, square, diagonal, scalar, and identity matrices, each with distinct properties.
• Matrix addition and subtraction require matrices of the same order, while matrix multiplication requires the number of columns of the first matrix to equal the number of rows of the second.
2. Transpose, Symmetric and Skew-Symmetric Matrices
Short Note:
• The transpose of a matrix A, denoted A', is obtained by interchanging its rows and columns; A is symmetric if A = A' and skew-symmetric if A' = -A.
• Any square matrix A can always be expressed as the sum of a symmetric matrix ½(A + A') and a skew-symmetric matrix ½(A - A').
3. Elementary Operations and Inverse of a Matrix
Short Note:
• Elementary row/column operations — swapping rows, multiplying a row by a nonzero scalar, or adding a multiple of one row to another — are used to simplify matrices and find inverses.
• A square matrix A has an inverse only if it is non-singular (|A| ≠ 0); the inverse can be computed using elementary operations or the adjoint method.
NCERT Textbook Exercise Questions
When are two matrices said to be equal?
Answer: Two matrices are said to be equal if and only if they have the same order (the same number of rows and columns) and all of their corresponding elements are identical. Even if two matrices contain the same numbers arranged differently, or have different orders, they are not considered equal.
If A is a square matrix, show that A + A' is always symmetric.
Answer: Consider the transpose of (A + A'), which equals A' + (A')' by the property of transposes distributing over addition. Since (A')' = A, this simplifies to A' + A, which is the same as the original expression A + A'. Since the matrix equals its own transpose, A + A' is symmetric for any square matrix A.
What condition must be satisfied for a matrix to have a multiplicative inverse?
Answer: A square matrix A has a multiplicative inverse if and only if it is non-singular, meaning its determinant |A| is not equal to zero. If |A| = 0, the matrix is called singular, and no inverse exists because the formula A⁻¹ = (1/|A|) adj A would involve division by zero.
What is the significance of elementary row operations in solving matrix problems?
Answer: Elementary row operations allow a matrix to be systematically simplified without altering the underlying system of equations it represents, making them useful for finding the rank of a matrix, solving systems of linear equations, and computing the inverse of a matrix. They include interchanging two rows, multiplying a row by a nonzero constant, and adding a multiple of one row to another.
