# Computers & Determinants

1. Jul 12, 2005

### amcavoy

How do computers evaluate determinants of large matrices? The cofactor method seems like it would be too time consuming. Does anyone know?

2. Jul 12, 2005

### matt grime

my guess would be row operations. even a naive method takes n-1 operations with n and multipkciations additions for the first column, n-2 operations with n-1 additions so approximately order n^4 operations to work out a diagonal form, then multiply the n thingst together, not much more work...so quite cheap really, and that's just the naive version.

3. Jul 12, 2005

### TOKAMAK

How would this fare computationally?

$$det(A) = \sum_{{i}_1, {i}_2,...,{i}_n = 1}^{N} \epsilon_{{i}_1, {i}_2,...,{i}_n} a_{{i}_1, 1} \cdot a_{{i}_2, 2} \cdot \cdot \cdot a_{{i}_n, N}$$

Cumbersome and inefficient for a computer algorithm?

4. Jul 12, 2005

### Hurkyl

Staff Emeritus
Well, how much work is it? You do n multiplications how many times?

5. Jul 12, 2005

### TOKAMAK

Looks like if you pick out all the distinct terms (the ones in the summation that aren't equal to zero) you get (N!). On top of that, you would need to do N multiplications each time (so N multiplications N! times). Yikes

Yeah didn't work that one out before

6. Jul 16, 2005

### TheGinkgoNut

7. Jul 16, 2005

### matt grime

I would think there are indeed many better ways than mine to find determinants. i would say mine is the best worst method or thee worst best method: the least complicated way of doing it that isn't naive. i've been told of other methods for speeding it up though i cant' recall them. of course when we think of a matrix we often think of something small with bige entries (ie integers) frequetly we need a clever method of finding it in the real world since we often have small entries, ie (numbers ivolved in the computation of) determinants so small that it may appear singular when it isn't.

8. Aug 28, 2005

### amcavoy

So row-reduction (to get 0-entries) would be more efficient? Or would that take a long time in itself?