What is naive Gaussian elimination?

Naive is the application of Gaussian elimination to solve systems of linear equations with the assumption that pivot values will never be zero.

Gaussian elimination attempts to convert a system of linear equations from a form like: ##color(white)("XXX")( (a_(1,1),a_(1,2),a_(1,3),"...",a_(1,n)),(a_(2,1),a_(2,2),a_(2,3),"...",a_(2,n)),(a_(3,1),a_(3,2),a_(3,3),"...",a_(3,n)),("...","...","...","...","..."),(a_(n,1),a_(n,2),a_(n,3),"...",a_(n,n)))xx((x_1),(x_2),(x_3),("..."),(x_n))=((c_1),(c_2),(c_3),("..."),(c_n))##

into a form like: ##color(white)("XXX")( (1,hata_(1,2),hata_(1,3),"...",hata_(1,n)),(0,1,hata_(2,3),"...",hata_(2,n)),(0,0,1,"...",hata_(3,n)),("...","...","...","...","..."),(0,0,0,"...",1))xx((x_1),(x_2),(x_3),("..."),(x_n))=((hatc_1),(hatc_2),(hatc_3),("..."),(hatc_n))##

A critical step in this process is the ability to divide row values by the value of a "pivot entry" (the value of an entry along the top-left to bottom-right of (a possibly modified) coefficient matrix.

Naive Gaussian Elimination assumes that this division will always be possible i.e. that the pivot value will never be zero. (Note, by the way, a pivot value close to but not necessarily equal to zero, can make the results unreliable when working with calculators or computers with limited accuracy).