Chapter 2.1, Problem 6P

Explanation of Solution

Proving that computing the matrix product AB requires $n^3$multiplications and $n^3-n^2$ additions:

When we multiply two $n\times n$ matrices $A$ and $B$, the resultant matrix will also be of the order of $n\times n$.

Therefore, $ij$ element of $AB=$ Scalar product of row $i$ of $A$ and column $j$ of $B$.

This will constitute $n\times n=n^2$ operations. Totally, $n$ such operations have to be done.

Therefore, total number of multiplication operations in multiplying two $n\times n$ matrices involve are given below:

$n^2\times n=n^3$

After this we must do $n^2$ addition operations in first row, then ${\left(n-1\right)}^2$ addition operations in second row and so on.

Therefore, the total number of addition operations is given below:

$n^2+{\left(n-1\right)}^2+\mathrm{}$