invertiblellä

Invertible refers to a property of mathematical objects, most commonly functions or matrices, indicating that an inverse operation exists. For a function, being invertible means that for every output, there is a unique input that produced it, allowing for a reverse mapping. If a function f maps x to y, its inverse function, often denoted as f⁻¹, maps y back to x. This inverse function effectively "undoes" the original function.

In the context of matrices, an invertible matrix is a square matrix that has a multiplicative inverse. This inverse matrix, when multiplied by the original matrix, results in the identity matrix. A square matrix is invertible if and only if its determinant is non-zero. The existence of an inverse for a matrix has significant implications in solving systems of linear equations, as it allows for direct computation of the solution.

The concept of invertibility is fundamental across various branches of mathematics, including algebra, calculus, and linear algebra. It is crucial for operations like solving equations, transforming data, and understanding the structure of mathematical systems. An object that is not invertible is called singular.