No, a matrix cannot have all zero entries and still be invertible. An invertible matrix, also known as a non-singular or a non-degenerate matrix, is a square matrix that has an inverse. In order for a square matrix to have an inverse, it must have a non-zero determinant.

  The determinant of a matrix is a scalar value that can be computed from its entries. If a matrix has all zero entries, then its determinant is always zero. This means that the matrix is not invertible.

  To understand why this is the case, consider the definition of matrix inverse. For a matrix A to be invertible, there must exist a matrix B such that the product AB is equal to the identity matrix I. In other words, if A is invertible, then there exists a matrix B such that AB = BA = I.

  If all entries of A are zero, then no matter what matrix B is chosen, the product AB will only consist of zero entries. Therefore, it is not possible to find a matrix B that satisfies the condition for invertibility, and the matrix A is not invertible.

  In summary, a matrix with all zero entries is always singular and not invertible.