Matrices

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 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.

Matrices can be classified based on their dimensions. A matrix with m rows and n columns is called an m-by-n matrix. For example, a 2-by-3 matrix has two rows and three columns. Special cases include square matrices, where the number of rows equals the number of columns (m = n), and vectors, which are matrices with only one row or one column.

Matrices can be added, subtracted, and multiplied. Addition and subtraction are performed element-wise, provided that the matrices have the same dimensions. Matrix multiplication is more complex and involves a dot product of rows and columns. The product of an m-by-n matrix and an n-by-p matrix is an m-by-p matrix.

Matrices can also be multiplied by scalars, which involves multiplying each element of the matrix by the scalar. This operation is known as scalar multiplication.

In addition to arithmetic operations, matrices can be transposed, which involves flipping the matrix over its diagonal, swapping rows with columns. The transpose of a matrix A is denoted by A^T.

Matrices are used to represent linear transformations, solve systems of linear equations, and perform various computations in computer graphics and machine learning. They are a powerful tool for organizing and manipulating data in a structured and efficient manner.