Expose both the parent matrix and its child matrix items on a saved item search filtered by Category.

1. Navigate to Lists > Search > Saved Searches > New.
2. Select Item from the New Saved Search list.
3. Under Criteria tab > Standard sub tab, set Use Expressions = True.
4. Add the following filters : 
        Display in Website is True
        (Category includes <specify the particular site category>  Or Parent : Category includes <specify the particular site category>)
5. Click the Results tab > Columns sub tab to add and/or remove column fields.
6. Give the search a unique name.
7  Click Save & Run.