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Encyclopedic overview
In linear algebra, an invertible matrix (non-singular, non-degenerate or regular) is a square matrix that has an inverse. In other words, if a matrix is invertible, it can be multiplied by its inverse matrix to yield the identity matrix. Invertible matrices are the same size as their inverse.
The inverse of a matrix represents the inverse operation, meaning if a matrix is applied to a particular vector, followed by applying the matrix's inverse, the result is the original vector.
Excerpted from Wikipedia’s “invertible matrix” article, available under the CC BY-SA 4.0 licence.