An invertible matrix is a square matrix that has an inverse. This means that when the matrix is multiplied by its inverse, the result is the identity matrix, which is a special matrix with ones on the diagonal and zeros elsewhere. Only square matrices can be invertible, and not all square matrices have inverses. A matrix is invertible if its determinant is non-zero. To find the inverse of a matrix, various methods can be used, such as the Gauss-Jordan elimination or the adjugate method. In practical applications, invertible matrices are essential in solving systems of linear equations, as they allow for unique solutions when the equations are represented in matrix form.