Let $A$ be an $n \times n$ matrix whose entries are all $0$ or $1$, and suppose $A^2 = 0$ (i.e., $A$ is nilpotent of order 2). What is the maximum number of $1$s that $A$ can contain? Prove your answer and give an explicit construction that achieves the maximum.

Maximum Ones in a Nilpotent Binary Matrix

Linear Algebra · Hard · Free problem

Let $A$ be an $n \times n$ matrix whose entries are all $0$ or

$, and suppose $A^2 = 0$ (i.e., $A$ is nilpotent of order 2). What is the maximum number of

$s that $A$ can contain? Prove your answer and give an explicit construction that achieves the maximum.