1.6. CROSS PRODUCTS
89
the row and column containing our entry. Thus
A
11
=
. .
.
. a
22
a
23
. a
32
a
33
A
12
=
. . .
a
21
. a
23
a
31
. a
33
A
13
=
.
. .
a
21
a
22
.
a
31
a
32
.
.
Now, the 3
×
3 determinant of
A
can be expressed as the
alternating
sum
of the
entries
of the first row times the
determinants of their minors
:
det
A
=
a
11
det
A
11
−
a
12
det
A
12
+
a
13
det
A
13
=
3
summationdisplay
j
=1
(
−
1)
1+
j
a
1
j
det
A
1
j
.
For future reference, the numbers multiplying the firstrow entries in the
formula above are called the
cofactors
of these entries: the cofactor of
a
1
j
is
cofactor(1
j
) := (
−
1)
1+
j
det
A
1
j
.
We shall see later that this formula usefully generalizes in several ways.
For now, though, we see that, once we have mastered this formula, we can
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 Fall '08
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 Calculus, Determinant, Howard Staunton, det A1j, a1j det A1j

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