This article describes searching for The Minimum Spanning Tree of the graph using the Weighted Adjacency Matrix. It describes the Weighted Adjacency Matrix as a new element, and shows how it could be used for searching for the Minimum Spanning Tree of the graph. This creates a new procedure for searching for the Minimum Spanning Tree of the graph which completes previously known algorithms of searching for the Minimum Spannig Tree. The proposed Weighted Adjacency Matrix could be used in similar issues in the field of the graph theory, where graphs with weighted edges are given. The procedure is shown on the attached example.