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I am not exactly getting the question, could anyone help me by an example?

(1) How many algebraic operations $(+,-,\times , /)$ are nessesary to multiply a $n\times n$ real matrix by a real $n$ touple vector?

(2) same question as above but to solve $Ax=b$ where $A$ is an upper triangular matrix having no element as $0$

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    For (1), the question is: "Let's say you know the entries of $A$ and $b$ and you want to write down the vector $Ab$. If your hand calculator charges you $\$1$ every time you press one of the aforementioned buttons, how much money will you pay?". And similarly for (2). – 2017-01-25
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    for multiplication and addition for each Ab I need $2n$ operation and I have $n$ entires so as whole $2n^2$ right? – 2017-01-25
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    It is possible to do much better than the naive $O(n^3)$ approach. See https://en.wikipedia.org/wiki/Strassen_algorithm. AFAIK, the actual minimum number of operations is not known. – 2017-01-25

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Suppose $AB = Q$

Each element in $Q$ is created when a row of $A$ multiplies by a column of $B.$

or $q_{i,j} = \sum_\limits{k = 1}^n a_{i,k}b_{k,j}$

For each element in $Q$ then there are $n$ multiplication operations and $(n-1)$ addition operations.

And there are $n^2$ elements in $Q.$

Can you apply similar logic for your second question?

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    Your answer is correct about the number of operations used by the usual naive algorithms, but it is known that one can do much better than that. – 2017-01-25