Linear Algebra

296 Chapter 4. Determinants
also does not give the same determinant as before the swap — again there is a
sign change. Trying a different 3×3 swapρ1↔ρ2 ⎛ ⎞
d e f
det( ⎝a b c⎠)
=dbi + ecg + fah− hcd − iae − gbf
g h i
also gives a change of sign.
Thus, row swaps appear to change the sign of a determinant. This modifies
our plan, but does not wreck it. We intend to decide nonsingularity by
considering only whether the determinant is zero, not by considering its sign.
Therefore, instead of expecting determinants to be entirely unaffected by row
operations, will look for them to change sign on a swap.
To finish, we compare det( ˆ T ) to det(T ) for the operation
T kρi
−→ ˆ T
of multiplying a row by a scalar k �= 0. One of the 2×2 cases is
� �
a b
det( )=a(kd) − (kc)b = k · (ad − bc)
kc kd
and the other case has the same result. Here is one 3×3 case
⎛
⎞
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)
and the other two are similar. These lead us to suspect that multiplying a row
by k multiplies the determinant by k. This fits with our modified plan because
we are asking only that the zeroness of the determinant be unchanged and we
are not focusing on the determinant’s sign or magnitude.
In summary, to develop the scheme for the formulas to compute determinants,
we look for determinant functions that remain unchanged under the
pivoting operation, that change sign on a row swap, and that rescale on the
rescaling of a row. In the next two subsections we will find that for each n such
a function exists and is unique.
For the next subsection, note that, as above, scalars come out of each row
without affecting other rows. For instance, in this equality
⎛ ⎞ ⎛ ⎞
3 3 9
1 1 3
det( ⎝2 1 1 ⎠) =3· det( ⎝2 1 1 ⎠)
5 10 −5
5 10 −5
the 3 isn’t factored out of all three rows, only out of the top row. The determinant
acts on each row of independently of the other rows. When we want to use
this property of determinants, we shall write the determinant as a function of
the rows: ‘det(�ρ1,�ρ2,...�ρn)’, instead of as ‘det(T )’ or ‘det(t1,1,...,tn,n)’. The
definition of the determinant that starts the next subsection is written in this
way.