Matrices and Determinants from "summary" of NCERT Class 10 Mathematics Solutions by JagranJosh
A matrix is a rectangular arrangement of numbers, symbols, or expressions in rows and columns. The numbers, symbols, or expressions in a matrix are called its elements. The horizontal lines in a matrix are called rows, and the vertical lines are called columns. The number of rows and columns in a matrix is called its order. For example, a matrix with m rows and n columns is said to be an m × n matrix. Matrices are used to represent data in a concise and organized way. They are widely used in various fields such as physics, computer science, economics, and engineering. Matrices allow us to perform operations such as addition, subtraction, multiplication, and division. The operations on matrices follow certain rules and properties, which make them useful for solving complex problems. Determinants are special numbers associated with square matrices. A determinant is a scalar value that can be calculated from the elements of a square matrix. The determinant of a matrix is denoted by |A|, where A is the matrix. Determinants play a crucial role in various mathematical applications, such as solving systems of linear equations, finding the inverse of a matrix, and calculating the area of a triangle. The determinant of a 2 × 2 matrix can be calculated using a simple formula. For a matrix A = [[a, b], [c, d]], the determinant |A| = ad - bc. The determinant of a 3 × 3 matrix can be calculated using the cofactor expansion method. The cofactor expansion method involves finding the determinants of smaller matrices called minors and multiplying them by the corresponding elements of the original matrix. Determinants have several properties that make them useful in matrix algebra. These properties include the linearity property, the product property, and the transpose property. The linearity property states that the determinant of a sum of matrices is equal to the sum of their determinants. The product property states that the determinant of a product of matrices is equal to the product of their determinants.- Matrices and determinants are fundamental concepts in mathematics that are used to represent and manipulate data efficiently. Matrices help us organize and analyze data, while determinants help us solve complex problems involving square matrices. Understanding matrices and determinants is essential for mastering various mathematical concepts and applications.
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