I'm having a problem with this rule in general. Apparently one can calculate the determinant by multiplying the cofactors and entries of any row or any column of a matrix. I have a negative that pops up. I'll take a 3X3 matrix for simplicity.(adsbygoogle = window.adsbygoogle || []).push({});

A=

|a b c|

|d e f|

|g h i|

calculating from the top row we have:

det(A)=a(ei-fh)-b(di-fg)+c(dh-eg) = aei-afh-bdi+bfg+cdh-ceg

calculating from the middle row we have

det(A)=d(bi-ch)-e(ai-cg)+f(ah-bg) = -aei+afh+bdi-bfg-cdh+ceg

The determinant seems to change by a factor of negative 1. This also seems to make sense from row operations. If for example, I swap the first and the second row to yield a matrix B, Then det(B)=-det(A). Yet if I calculate using entries and cofactors corresponding to the second row of B and the first row of A, the cofactor expansion of the two determinants will be identical. Though I'm told that we can start from any row or column, what did I miss? Thanks all.

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Determinants from any row or column

**Physics Forums | Science Articles, Homework Help, Discussion**