Class 12 Maths - CBSE
Matrices
The Matrices chapter in Class 12 CBSE Mathematics introduces a rectangular array of numbers that serves as a powerful tool for solving systems of linear equations in business, engineering, and science. This chapter covers the definition of a matrix, its various types, operations like addition, scalar multiplication, and multiplication, along with the concept of transpose, symmetric, and skew-symmetric matrices. A major focus is placed on invertible matrices and finding the inverse of a matrix using elementary transformations or the adjoint method. For board exams, this chapter is fundamental because it connects directly with Determinants and forms the backbone of the 6-mark long-answer question on solving matrix equations.
Start Learning FreeKey Concepts
Order of a Matrix
A matrix with m rows and n columns is said to be of the order m x n, where the total number of elements is the product mn.
Matrix Multiplication
Two matrices A and B can be multiplied only when the number of columns in A is equal to the number of rows in B. Matrix multiplication is not commutative in general.
Transpose of a Matrix
The transpose of a matrix is obtained by interchanging its rows and columns, denoted by A', with properties like (A')' = A and (AB)' = B'A'.
Symmetric and Skew-Symmetric Matrices
A square matrix A is symmetric if A' = A and skew-symmetric if A' = -A. Every square matrix can be uniquely expressed as the sum of a symmetric and a skew-symmetric matrix.
Inverse of a Matrix
A square matrix A of order n is invertible if there exists a square matrix B of the same order such that AB = BA = I, where B is denoted as A^(-1) = (adj A) / |A|.
Important Formulas
Board Exam Info
In the CBSE Class 12 Mathematics board exam, Matrices (along with Determinants) typically carries around 10 to 12 marks. Common question types include 1-mark multiple-choice questions on types of matrices, 2-mark or 3-mark short-answer questions on matrix operations and properties of transpose/symmetric matrices, and a crucial 5-mark or 6-mark long-answer question based on solving a system of linear equations using the matrix method (A^(-1)B).
Frequently Asked Questions
Is matrix multiplication commutative?
No, matrix multiplication is not commutative in general, meaning AB is rarely equal to BA, even when both products are defined.
How do I know if a matrix is invertible?
A square matrix is invertible if and only if it is a non-singular matrix, which means its determinant is not equal to zero (|A| != 0).
What is the difference between symmetric and skew-symmetric matrices?
For a symmetric matrix, the transpose equals the matrix itself (A' = A), whereas for a skew-symmetric matrix, the transpose equals the negative of the matrix (A' = -A) and all its diagonal elements must be zero.
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