Yes, a matrix can have a negative determinant.

  The determinant of a matrix is a scalar value that represents certain properties of the matrix. It is calculated using specific operations on the elements of the matrix. The sign of the determinant, positive or negative, indicates whether the matrix is invertible or whether the matrix reflects or flips the orientation of vector spaces, among other things.

  For example, consider a 2x2 matrix:

  | a b |

  | c d |

  The determinant of this matrix is calculated as ad - bc. If the determinant is negative, it means that the matrix has a "negative" impact on space, flipping its orientation. If the determinant is positive, it means that the matrix preserves the orientation of space.

  This concept extends to matrices of larger sizes as well. A square matrix with a negative determinant will have a similar effect on space, changing its orientation. It is important to note that the magnitude of the determinant does not affect the sign. It is possible to have a matrix with a very large negative determinant or a very small negative determinant, depending on the specific values in the matrix.

  In summary, a matrix can indeed have a negative determinant, and this signifies a change in orientation in the vector space it operates on.