symmetric $(v,k,\lambda)$-design

Question

I need help to prove the lemma, knowing the following corollary about the symmetric $(v,k,\lambda)$-designs.

Corollary. Let $A=\begin{bmatrix} A(i,j) \end{bmatrix}$ be the associated matrix of some $(v,k,\lambda)$-design symmetric. Then,

1. $ A^TA =(k-\lambda)I_v+\lambda J_v;$
2. $AA^T =(k-\lambda)I_v+\lambda J_v;$
3. $J_vA=kJ_v;$
4. $AJ_v=kJ_v.$

Lemma. Let $A=\begin{bmatrix} A(i,j) \end{bmatrix}$ be a $\{0,1\}$-valued non-singular matrix of order $v$. Then, Equation $1.$ is true iff Equation $2.$ is true. Moreovoer, the $A$ is an associate matrix of some $(v,k,\lambda)$-design symmetric iff $1.$ (or $2.$) is satisfied.

Thanks for your help..

Answer 1

You need to show that $A$ commutes with $A^T$. If 1 holds, $A^T=(k-\lambda)A^{-1}+A^{-1}J$. As $A$ commutes with $A^{-1}$ and with $J$, it commutes with $A^{-1}J$ and so with $A^T$.