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.

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# Homework Help: Determinants from any row or column

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