# Computers & Determinants

## Main Question or Discussion Point

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

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matt grime
Homework Helper
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.

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?

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

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

matt grime