matriks

Matriks, in mathematics, is a rectangular array of numbers, symbols, or expressions arranged in rows and columns. A matriks A has size m by n, with entries a_ij denoting the element in the i-th row and j-th column. Matriks provide a compact way to represent linear transformations and systems of linear equations.

Operations on matriks include addition and subtraction, which require matching dimensions, and scalar multiplication, which multiplies every entry by a scalar. Matrix multiplication is defined when the number of columns of the first matriks equals the number of rows of the second; the product is an m by p matriks with entries c_ij = sum_k a_ik b_kj.

Key concepts also include the transpose A^T, which flips rows and columns. For square matriks, the determinant det(A) is a scalar indicating properties like invertibility, with A invertible if det(A) ≠ 0 and A·A^−1 = I. The inverse exists only for square, full-rank matriks. The rank of a matriks is the maximum number of linearly independent rows or columns, and equals the dimension of its image.

Special types of matriks include diagonal matriks, which have nonzero entries only on the main diagonal; the identity matriks I, with ones on the diagonal and zeros elsewhere; and the zero matriks 0. A matriks is symmetric if A^T = A, and skew-symmetric if A^T = −A. Orthogonal matriks satisfy A^T A = A A^T = I, with A^−1 = A^T for real matrices.

Applications of matriks span solving linear systems, describing linear transformations in geometry and computer graphics, and supporting data analysis, statistics, and machine learning. They underpin eigenvalue problems, decomposition methods, and many algorithms across science and engineering. The concept arose in 19th-century mathematics, with Sylvester and Cayley contributing to matrix notation and theory.