Еhis is a task on the topic "bilinear and quadratic forms".
Problem: For which number $n$ there exists a square matrix of order $n$ with elements 0, 1 such that its square is a matrix of only ones.
Well, when I tried find the solution, I came to fact that $n$ only can be 1.
Let $J_n$ be the square matrix of ones (all entries are a one).
I tried to find new basis where quadratic form with matrix $J_n$ where $J_n$ became diagonal matrix, but can't find $n$ linearly independent vectors for this basis. Because of this I suppose that $n$ only can be equal 1.
Also tried to make linear system of equations with $n^2$ rows and again came to a standstill.