I am reading my Linear Algebra book, and one of the exercises is asking me to find the entry (2,2) of a product of 3 3x3 matrices ABC without calculating the full product.
My current approach would be to:
- Calculate (1,2),(2,1),(2,2),(2,3),(3,2) of AB
- Then use that result to calculate (2,2) of (AB)C
Is this the fastest approach?
Hint: If $e_2 = (0,1,0)^T$, then the $2,2$ entry of our product can be written as $$ e_2^T (ABC)e_2 = [(e_2^TA)B](Ce_2) $$