Chapter 2.2, Problem 21E

Show that the elimination method of computing the value of the determinant of an $n\times n$ matrix involves $\left[n\left(n-1\right)\left(2n-1\right)\right]/6$ additions and $\left[\left(n-1\right)\left(n^2+n+3\right)\right]/3$ multiplications and divisions. Hint: At the ith step of the reduction process, it takes $n-i$ divisions to calculate the multiples of the ith row that are w be subtracted from the remaining rows below the pivot. We must then calculate new values for the ${\left(n-i\right)}^2$ entries in rows $i+1$ through n and columns $i+1$ through n.