What does it mean for a matrix to be invertible?

A matrix is considered invertible if there exists another matrix, often referred to as its inverse, such that when the two matrices are multiplied together, the result is the identity matrix. The identity matrix is a special square matrix that has ones on the diagonal and zeros elsewhere, serving as the multiplicative identity in matrix multiplication. This relationship implies that for a matrix ( A ), if there is a matrix ( B ) such that ( AB = BA = I ) (where ( I ) is the identity matrix), then the matrix ( A ) is invertible, and ( B ) is its inverse. This property is fundamental in linear algebra, as it demonstrates that the transformation represented by ( A ) can be reversed by ( B ), leading to a unique solution in systems of linear equations.

The other options reflect common misconceptions or other properties that may not directly relate to the definition of invertibility. For instance, a nonzero determinant indicates that a matrix is invertible, but not having a zero determinant or the presence of positive eigenvalues do not provide a sufficient or necessary condition independently.