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Engineering Calculus Notes 101

# Engineering Calculus Notes 101 - 89 1.6 CROSS PRODUCTS the...

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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 first-row 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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