tersinir

Tersinir is a Turkish term used in mathematics, often in the context of linear algebra, to describe an object that has an inverse. In this sense, a square matrix A is tersinir if there exists a matrix B such that AB = BA = I, where I is the identity matrix. B is called the inverse of A and is denoted A−1.

Several equivalent conditions characterize tersinir matrices. A matrix A is tersinir if and only if its determinant is nonzero (det(A) ≠ 0). Equivalently, A has full rank (rank(A) = n for an n×n matrix); the homogeneous system Ax = 0 has only the trivial solution; the columns of A form a basis for the space; A can be row-reduced to the identity matrix; and 0 is not an eigenvalue of A.

If A is tersinir, the inverse A−1 is unique. Inverses obey several useful rules, such as (AB)−1 = B−1A−1 for invertible A and B, (A^T)−1 = (A−1)^T, and (A^k)−1 = (A−1)^k for any positive integer k. The inverse also provides a direct method to solve linear systems: Ax = b has a unique solution x = A−1b whenever A is tersinir.

Examples help illustrate the concept. The 2×2 matrix A = [[1, 2], [3, 4]] has determinant det(A) = 1·4 − 2·3 = −2 ≠ 0, so A is tersinir, and its inverse is A−1 = (1/−2) [[4, −2], [−3, 1]] = [[−2, 1], [3/2, −1/2]]. A matrix with det 0, such as [[1, 2], [2, 4]], is singular and not tersinir; it has no inverse.