Question about symmetric block design and Hadamard matrix

Question

I stock in middle of proving that if $A$ is matrix of symmetric block design and $B = 2A - J$ that $J$ is ones matrix then B is a Hadamard matrix if and only if $v = 4(k-\lambda)$. I need to prove that $$ \begin{equation*} -2(JA)^T - 2JA + J^2 = (-4k + v)J \end{equation*} $$ is there any helpful hint?

Answer 1

The adjacency matrix of a symmetric block design satisfies $AA^T=(r-\lambda)I+\lambda J$ and $JA=AJ=kJ$. An Hadamard matrix $H$ satisfies $HH^T=nI$. So now let us count $BB^T$: $$ BB^T=(2A-J)(2A-J)^T=4AA^T-2AJ-2JA+JJ^T=4((r-\lambda)I+\lambda J)-4kJ+vJ. $$ Thus, $BB^T=4(r-\lambda)I+[4(\lambda-k)+v]J$. Therefore $B$ is Hadamard if the coefficient of $J$ is zero, hence $v=4(k-\lambda)$.