Determinants

Using properties of determinants and applying row and column operations, we find that the determinant simplifies to .

The matrix is a special matrix known as a "min matrix," where each element is the minimum of its row and column indices. It is known that the determinant of such a matrix, regardless of the size, equals 1. This is a result derived from properties of the Vandermonde-like matrices.

To evaluate the determinant, use cofactor expansion along the first row. After calculation, we find that the determinant does not simplify to any of the options listed. Therefore, the correct answer is (d).

Step 1: Recall the determinant property.
For any square matrix of order , if each element of the matrix is multiplied by a constant , then the determinant is multiplied by .
Step 2: Apply the formula.
Here, the order of matrix is , and the scalar multiplier is . Hence,
Step 3: Substitute the given value.

Step 4: Conclusion.
Therefore, the value of the determinant is .