Linear Algebra, 2020a

328 Chapter Four. Determinants
2×2 row swap ρ 1 ↔ ρ 2 does not yield ad − bc.
( )
c d
det( )=bc − ad
a b
And this ρ 1 ↔ ρ 3 swap inside of a 3×3 matrix
⎛ ⎞
g h i
⎜ ⎟
det( ⎝d e f⎠) =gec + hfa + idb − bfg − cdh − aei
a b c
also does not give the same determinant as before the swap since again there is
a sign change. Trying a different 3×3 swap ρ 1 ↔ ρ 2
⎛ ⎞
d e f
⎜ ⎟
det( ⎝a b c⎠) =dbi + ecg + fah − hcd − iae − gbf
g h i
also gives a change of sign.
So row swaps appear in this experiment to change the sign of a determinant.
This does not wreck our plan entirely. We hope to decide nonsingularity by
considering only whether the formula gives zero, not by considering its sign.
Therefore, instead of expecting determinant formulas to be entirely unaffected
by row operations we modify our plan so that on a swap they will change sign.
Obviously we finish by comparing det(ˆT) with det(T) for the operation of
multiplying a row by a scalar. This
( )
a b
det( )=a(kd)−(kc)b = k · (ad − bc)
kc kd
ends with the entire determinant multiplied by k, and the other 2×2 case has
the same result. This 3×3 case ends the same way
⎛
⎞
a b c
⎜
⎟
det( ⎝ d e f⎠) =ae(ki)+bf(kg)+cd(kh)
kg kh ki −(kh)fa −(ki)db −(kg)ec
= k · (aei + bfg + cdh − hfa − idb − gec)
as do the other two 3×3 cases. These make us suspect that multiplying a row
by k multiplies the determinant by k. As before, this modifies our plan but does
not wreck it. We are asking only that the zero-ness of the determinant formula
be unchanged, not focusing on the its sign or magnitude.
So in this exploration our plan got modified in some inessential ways and is
now: we will look for n×n determinant functions that remain unchanged under