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A matrix is a rectangular array or table of numbers, symbols, or expressions, arranged in rows and columns. Matrices are fundamental in linear algebra and have wide applications in various fields such as physics, engineering, computer science, and economics. They are typically denoted by uppercase letters, such as A, B, or C.

The size of a matrix is defined by the number of rows and columns it contains. A matrix with m rows and n columns is called an m x n matrix. For example, a 2 x 3 matrix has two rows and three columns. Matrices can be square, meaning they have the same number of rows and columns (e.g., 3 x 3), or rectangular, meaning they have a different number of rows and columns (e.g., 2 x 3).

Matrices can be used to represent systems of linear equations, transformations in geometry, and data sets. They can be added, subtracted, and multiplied, following specific rules of matrix algebra. The transpose of a matrix is obtained by swapping its rows with columns, and the determinant of a square matrix provides information about the matrix's invertibility and geometric properties.

In computer science, matrices are used to represent images, perform operations on vectors, and solve systems of linear equations efficiently. In physics, matrices are employed to describe quantum states and operators. In economics, matrices are used to model input-output systems and analyze economic data. The study of matrices and their properties is a rich area of mathematics with numerous practical applications.