Calculating determinant by cofactors

[Bob]

Show that evaluating the determinant of an n*n matrix by cofactors involves (n!-1) additions and $$\sum^{n-1}_{k=1}n!/k!$$multiplications.

What does it mean? how to do it? Help!

 

[matt grime]

Induction. You're trying to count the number of basic operations in expanding this (a necessary technique if one wants to estimate how long it will take on a computer). You can do this inductively since expanding by cofactors writes an nxn determinant in terms of n lots of (n-1)x(n-1) determinants.

 

[HallsofIvy]

As and example, to find the determinant
$$\left|\begin{array}{cc}a && b \\ c && d\end{array}\right|= ad- bc$$
You must do (2!)- 1= 1 addition (ad+ (-bc)) and
$$\sum^{2-1}_{k=1}2!/k= \frac{2!}{1}= 2$$
multiplications, ad and bc.

 

[Bob]

thank you!