To understand the given question, let's first define what a diagonal matrix is.

Diagonal Matrix:
A diagonal matrix is a square matrix in which all the elements outside the main diagonal (the diagonal from the top left to the bottom right) are zero. In other words, all the elements that are not on the main diagonal are zero.

Given that D = diag (d1, d2,.....,dn), we can represent the diagonal matrix D as follows:

D = | d1 0 0 ... 0 |
| 0 d2 0 ... 0 |
| 0 0 d3 ... 0 |
| 0 0 0 ... dn |

Now, let's analyze the options provided and determine the correct answer.

Option (a): Dn
If we multiply the diagonal matrix D by n, each element of the matrix will be multiplied by n. Therefore, the resulting matrix, Dn, will be:

Dn = | nd1 0 0 ... 0 |
| 0 nd2 0 ... 0 |
| 0 0 nd3 ... 0 |
| 0 0 0 ... ndn |

Option (b): diag (d1n-1, d2n-1,.....dnn-1)
In this option, each element of the original diagonal matrix D is raised to the power of n-1. Therefore, the resulting diagonal matrix will be:

diag (d1n-1, d2n-1,.....dnn-1) = | d1n-1 0 0 ... 0 |
| 0 d2n-1 0 ... 0 |
| 0 0 d3n-1 ... 0 |
| 0 0 0 ... dnn-1 |

Option (c): diag( d1n,d2n,......dnn)
In this option, each element of the original diagonal matrix D is raised to the power of n. Therefore, the resulting diagonal matrix will be:

diag( d1n,d2n,......dnn) = | d1n 0 0 ... 0 |
| 0 d2n 0 ... 0 |
| 0 0 d3n ... 0 |
| 0 0 0 ... dnn |

Option (d): None of these
Since options (a) and (b) are incorrect, and option (c) is the correct answer, option (d) is not applicable.

Therefore, the correct answer is option (c), which represents the diagonal matrix diag( d1n,d2n,......dnn).