Help needed with VLOOKUP OR INDEX

We are working on a data sheet where we need to find the lowest price sold for each product and to the customer it was sold:
We have Products e.g. Product A, Product B, Product C,..... and this product is sold for different prices to different business customers.


When we use the formula it just gives us the lowest value of the table and not of that particular product. We know that we are missing something as the formula just work like MIN formula. I have attached the excel sheet with the formulas:

Product Price Customer
Product A 7 Customer 1
Product C 8 Customer 1
Product A 8 Customer 2
Product B 5 Customer 1
Product A 12 Customer 3
Product B 3 Customer 2
Product D 5 Customer 3
Product C 9 Customer 2
Product B 6 Customer 3
Product D 6 Customer 1


Customer 1 3 .=INDEX(MIN(B2:B11),MATCH(A14,A2:A11,0))
Customer 2
Customer 3
Customer 4

Hi,
Thank you for looking into this and helping us, I have redone the table to make it more clear.

Product Price Customer
Product A 7 Customer 1
Product C 8 Customer 1
Product A 8 Customer 2
Product B 5 Customer 1
Product A 12 Customer 3
Product B 3 Customer 2
Product D 5 Customer 3
Product C 9 Customer 2
Product B 6 Customer 3
Product D 6 Customer 1


Product A 3 .=INDEX(MIN($B$2:$B$11),MATCH(A14,$A$2:$A$11,0))
Product B #REF!
Product C #REF!
Product D #REF!


As we heave 'Product A' 3 times in above table sold for 3 different prices
The formula should show us the lowest price for 'Product A' but it is showing us the lowest price of all Products.
also when I drag down the formula it give me above error

Thank you very much Glenn Kennedy, both formula works and solved the issue.

Best Regards,