3'1 Definition

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Chapter 3 Determinants

• 3.1 Definition
• 3.2 Properties of Determinants
• 3.3 Cofactor Expansion
• 3.4 Inverse of a Matrix
• 3.5 Other Applications of Determinants
• 3.6 Computing Determinants

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3.3 Cofactor Expansion

• Comment - To this point, there has been only two
  ways to calculate the determinant
• Definition
• Reduction to triangular form
• There are other ways of calculating the
  determinant that can be useful for calculations
  and for proofs

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3.3 Cofactor Expansion

• Defn - Let A aij be an n x n matrix. Let
  Mpq be the (n 1) x (n 1) submatrix of A
  obtained by deleting the p th row and q th column
  of A. The determinant det(Mpq) is called the
  minor of apq
• Defn - Let A aij be an n x n matrix. The
  cofactor Apq of apq is defined as Apq (1) pq
  det( Mpq )

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Cofactor Expansion
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3.3 Cofactor Expansion

• Examine pattern of signs of term ( 1 ) pq
• When using cofactors, dont have to evaluate
  ( 1 ) pq Just remember checkered pattern

n 4
n 3
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3.3 Cofactor Expansion

• Theorem - Let A aij be an n x n matrix.
  Then
• det( A ) ai 1A i 1 ai 2A i 2 L ain A in
• expansion of det(A) with respect to row i
• det( A ) a1j A 1j a2j A 2j L anj A nj
• expansion of det(A) with respect to column j

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3.3 Cofactor Expansion

• Determinant of a 3 x 3 matrix
• Definition
• Reorganize expression with respect to first row

det( A ) a11a22a33 a12a23a31 a13a21a32
a11a23a32 a12a21a33 a13a22a31
det( A ) a11 ( a22a33 a23a32 )
a12 ( a23a31 a21a33 )
a13 ( a21a32 a22a31 )
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3.3 Cofactor Expansion

• Rewrite in terms of the cofactors

det(A) a11(a22a33 a23a32) a12(a23a31
a21a33) a13(a21a32 a22a31)
det(A) a11(a22a33 a23a32) a12(a21a33
a23a31) a13(a21a32 a22a31)
So det(A) a11 A11 a12 A12 a13 A13
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3.3 Cofactor Expansion

• Determinant of a 3 x 3 matrix
• Definition
• Reorganize expression with respect to first
  column

det( A ) a11a22a33 a12a23a31 a13a21a32
a11a23a32 a12a21a33 a13a22a31
det( A ) a11 ( a22a33 a23a32 )
a21 ( a23a31 a21a33 )
a13 ( a21a32 a22a31 )
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Cofactor Expansion

• Regroup det(A) with respect to the first column

det(A) a11(a22a33 a23a32) a21(a13 a32
a12a33) a31( a12a23 a13a22)
det(A) a11(a22a33 a23a32) a21(a12a33 a13
a32) a31( a12a23 a13a22)
So det( A ) a11 A11 a21 A21 a31 A31
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3.3 Cofactor Expansion

• Evaluate
• Pick row or column with large number of zeros,
  such as column 2

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3.3 Cofactor Expansion