Rank of a Matrix:
The rank of a matrix is defined as the maximum number of linearly independent rows or columns in the matrix. It determines the dimension of the vector space spanned by the rows or columns of the matrix.

Singular Matrix:
A square matrix is said to be singular if its determinant is zero. In other words, if a matrix is singular, it does not have an inverse.

Rank of a Singular Matrix of Order 4:
To find the rank of a singular matrix of order 4, we need to determine the maximum number of linearly independent rows or columns it can have.

The rank of a matrix is always less than or equal to the minimum of the number of rows and columns. In this case, the matrix is of order 4, which means it has 4 rows and 4 columns.

Explanation:
Since the rank of a matrix is always less than or equal to the minimum of the number of rows and columns, the rank of a singular matrix of order 4 can be at most 4. Therefore, the correct answer is option 'd'.

To see why the rank cannot be greater than 4, consider the fact that a matrix with rank greater than 4 would have more linearly independent rows or columns than the number of rows or columns it has. This would imply that the determinant of the matrix is non-zero, contradicting the assumption that the matrix is singular.

Example:
Consider the following matrix A:
A = [1 0 0 0]
[0 0 1 0]
[0 1 0 0]
[0 0 0 0]

This matrix is a singular matrix of order 4 since its determinant is zero. The rank of matrix A is 2 because there are only two linearly independent rows or columns (the first and third rows).

Conclusion:
The rank of a singular matrix of order 4 can be at most 4. This means that the maximum number of linearly independent rows or columns in the matrix is 4.