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Unformatted text preview: 89 1.6. CROSS PRODUCTS
the row and column containing our entry. Thus ..
A11 = . a22 a23 . a32 a33 ...
A12 = a21 . a23 a31 . a33 .
A13 = a21 a22 . .
a31 a32 . 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 = a11 det A11 − a12 det A12 + a13 det A13
j =1 (−1)1+j a1j det A1j . For future reference, the numbers multiplying the ﬁrst-row entries in the
formula above are called the cofactors of these entries: the cofactor of a1j
cofactor(1j ) := (−1)1+j det A1j .
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
express the calculation of the cross product as
− ×− =
v1 v2 v3
w1 w2 w3 where
− =v − +v − +v −
− = w − +w − +w −.
2 Exercises for § 1.6
Practice problems: ...
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