maatrikli

Maatrikli is a rectangular array of numbers, symbols, or expressions arranged in rows and columns. In mathematics, maatriklid are used to represent linear transformations, systems of linear equations, and data. A maatrikli with m rows and n columns is called an m-by-n maatrikli and is often denoted A ∈ F^{m×n}, where F is a field such as the real or complex numbers.

Operations on maatriklid include addition and subtraction (element-wise) for matrices of the same size, and multiplication by a scalar. Matrix multiplication is defined for A of size m×n and B of size n×p, producing an m×p maatrikli AB, where each entry is a sum of products of corresponding entries. The transpose A^T interchanges rows and columns, yielding an n×m maatrikli. If a maatrikli is square (n×n) and invertible, it has an inverse A^{-1} satisfying AA^{-1} = A^{-1}A = I_n. The determinant det(A) provides a scalar criterion for invertibility in square matrices, and the rank of A measures the dimension of its linear image.

Common types include diagonal matrices (nonzero entries only on the main diagonal), identity matrices, symmetric matrices (A^T = A), orthogonal matrices (A^T A = I), and triangular matrices (upper or lower). Sparse maatriklid contain mostly zero entries and are common in large-scale computations.

In computing, maatriklid are stored as two-dimensional arrays in programming languages and manipulated by linear algebra libraries. They are used to solve systems of equations, perform coordinate transformations in computer graphics, analyze data in statistics and machine learning, and model physical, economic, and network systems.

See also linear algebra, vectors, linear transformation, determinant, eigenvalues.