Question

Let \( M \) be any square matrix of arbitrary order \( n \) such that \( M^2 = 0 \) and the nullity of \( M \) is 6. Then the maximum possible value of \( n \) (in integer) is ________

Hint:

For a nilpotent matrix \( M \) with \( M^2 = 0 \), the rank is strictly less than \( n \) and the nullity is given as \( n - \text{rank}(M) \).

Answer 1

Correct Answer: 12

Given that \( M^2 = 0 \), this implies that \( M \) is a nilpotent matrix. For a nilpotent matrix \( M \), the rank-nullity theorem states that: \[ \text{rank}(M) + \text{nullity}(M) = n, \] where \( n \) is the order of the matrix. Since the nullity of \( M \) is given as 6, we can use this information to find the maximum possible rank of \( M \). The rank of a nilpotent matrix is always less than or equal to \( n - 1 \), and since \( M^2 = 0 \), the rank must be at most \( n - 1 \). Therefore, the maximum possible value of \( n \) occurs when the rank is as large as possible while still satisfying \( \text{rank}(M) + 6 = n \). This gives the maximum value of \( n \) as 8. Thus, the maximum possible value of \( n \) is \( \boxed{8} \).