Grothendieck published an extraordinary paper entitled ”Resume de la theorie metrique des produits tensoriels topologiques” 
 in 1953. The main result of this paper is the inequality which is commonly known as Grothendieck Inequality. Following Kirivine, 
 in this article, we give the proof of Grothendieck Inequality. We reformulate it in different forms. We also investigate the 
 famous Grothendieck constant KG. The Grothendieck constant was achieved by taking supremum over a special class of matrices.But 
 our attempt will be to investigate it, considering a smaller class of matrices, namely only the positive definite matrices in 
 this class.