First, I'd like to mention that it would be great if we could get more information regarding the matrices. Like, are they only containing numbers, or also strings?
Something you could look into is the calculation of similarities done in the field of Recommender Systems. These similarity measures are often used to recommend media content to a user depending on similar media (often based on metadata stored in a dataframe/matrix).
One popular metric for checking how similar matrices are is the Pearson Correlation metric. In this link you get further information about how to calculate it.
For clustering, you could consider using the hierarchical agglomerative clustering method with the Pearson Correlation as the criterion (see link for some description in the SO answer).