Question-42

How many orthogonal matrices of order \(\displaystyle n\) exist that are made up of only zeros and ones?

• \(n\)
• \(n^2\)
• \(n!\)
• \(n^n\)

Hint

Permute

Answer

• \(n\)
• \(n^2\)
• \(n!\)
• \(n^n\)

Solution

Since the columns are orthonormal, given any two rows, zeros and ones should not appear in identical locations. Additionally, since each column/row is unit norm, there is exactly one position with a one while the rest are zeros, in each column/row. We can see that all such matrices can be obtained by permuting the columns/rows of the identity matrix. Such matrices are termed permutation matrices. Therefore, the required count is \(\displaystyle n!\)