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c4s2 - 4.2 The Determinant of a Square Matrix 1 Chapter 4...

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4.2 The Determinant of a Square Matrix 1 Chapter 4. Determinants 4.2 The Determinant of a Square Matrix Definition. The minor matrix A ij of an n × n matrix A is the ( n 1) × ( n 1) matrix obtained from it by eliminating the i th row and the j th column. Example. Find A 11 , A 12 , and A 13 for A = a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 . Definition. The determinant of A ij times ( 1) i + j is the cofactor of entry a ij in A , denoted A ij .
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4.2 The Determinant of a Square Matrix 2 Example. We can write determinants of 3 × 3 matrices in terms of cofactors: det( A ) = a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 = a 11 | A 11 | − a 12 | A 12 | + a 13 | A 13 | = a 11 a 11 + a 12 a 12 + a 13 a 13 . Note. The following definition is recursive . For example, in order to process the definition for n = 4 you must process the definition for n = 3, n = 2, and n = 1. Definition 4.1. The determinant of a 1 × 1 matrix is its single entry. Let n > 1 and assume the determinants of order less than n have been defined. Let A = [ a ij ] be an n × n matrix. The
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