True or False: Let $A,B \in \mathbb{R}^{d \times d}$ be two matrices. Computing the product $AB \in \mathbb{R}^{d \times d}$ takes roughly $d^2$ floating point operations.
I'm having a hard time computing floating point operations. I believe the above is false but I am not sure how do you show it. How do you compute FLOPS?
HINT
Well, the naive way of computing the product of two matrices $C = AB$ defines $$ c_{ij} = \sum_{k=1}^d a_{ik} b_{kj}. $$
Multiply the answers from (1) and (2) and you get the number of floating point operations needed.
Can you now complete this?
UPDATE
Following your comments, you (almost) got it correctly. Note that $$ c_{ij} = a_{i1}b_{1k} + a_{i2}b_{2k} + \ldots + a_{d1}b_{dk}, $$ and each term in the sum needs $1$ multiplication to be computed, which is $d$ multiplications total, as you note. Now there will be $d$ numbers to add, but that requires $d-1$ multiplications (it takes $1$ operation to add $2$ numbers, $2$ operations to add $3$ numbers, etc.).
Overall, computing one entry is $d+(d-1) =2d-1$ operations, and we have $d^2$ entries in the matrix, so a total of $d^2(2d-1) =\Theta(d^3)$ calculations.