A Linearity Property of Determinants
On. p. 173 is a property of determinants that I didn't mention in lecture, assuming you'd
pick up on it in reading Section 3.2. That property is useful for at least one WebWork
problem that a couple of people have asked about.
Suppose E œ Ò+" ÞÞÞ +4 ÞÞÞ +8 Ó is an 8 ‚ 8 matrix with columns +" , ÞÞÞ +4 , ÞÞÞ +8
Supppose one column, say column +4 , is replaced by a variable column B from ‘8 Þ
Let X ÐBÑ œ det Ò+" ÞÞÞ B ÞÞÞ +8 Ó . X ÐBÑ is a number so B È X ÐBÑ is a function from ‘8
to ‘" Þ
X is a linear transformation. this means that
i) X Ð- BÑ œ -X ÐBÑ for all B in ‘8 and all scalars -ß and
ii) X ÐB CÑ œ X ÐBÑ X ÐCÑ for all Bß C in ‘8
i) is true becuase of a property we already know about how factoring a
number out of a column affects the determinant:
X Ð- BÑ œ det Ò+" ÞÞÞ - B ÞÞÞ +8 Ó œ - det Ò+" ÞÞÞ B ÞÞÞ +8 Ó œ -X ÐBÑ
ii) is checked by expanding the determinants for X ÐB CÑ, X ÐBÑ, and
X ÐCÑ down the 4th column (see the $ ‚ $ example in Exercise 43 of
Section 3.2.
To illustrate with a $ ‚ $ example.
Suppose E œ
Ô"
%
Õ(
#
&
)
$×
' ß and replace (say) column 2 with a veriable vector B
*Ø
from ‘$ :
Ô"
X ÐBÑ œ det %
Õ(
B"
B#
B$
$×
' Þ
*Ø
Ô"
Since X is linear, X ÐB CÑ œ det %
Õ(
Ô"
œ det %
Õ(
B" C"
B# C#
B$ C$
B"
B#
B$
$×
'
*Ø
$×
Ô"
' det %
Õ(
*Ø
C"
C#
C$
$×
' œ X Ð B Ñ X ÐC Ñ
*Ø
Read in reverse order, this says that if we have two matrices that are identical except for 1
column, then in computing the determinants
Ô " B"
det % B#
Õ ( B$
Å
$×
Ô " C"
' det % C#
Õ ( C$
*Ø
Å
Å
two columns the same
$×
Ô " B" C " $ ×
' œ det % B# C# '
Õ ( B$ C $ * Ø
*Ø
Å
Å
we can add together the two different
columns
note: we are not adding column 1's or
column 3's together