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cofactor


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Let A be the square matrix [aij]. The cofactor, Aij, of the entry aij is equal to (−1)i+j times the determinant of the matrix obtained by deleting the i-th row and j-th column of A. If A is the 3 × 3 matrix shown, the factor (−1)i+j has the effect of introducing a+or−sign according to the pattern on the right: So, for example, for a 2 × 2 matrix, the pattern is:So, the cofactor of a equals d, the cofactor of b equals−c, and so on. The following properties hold, for an n × n matrix A: (i) The expression ai1Ai1+ai2+⋯+ ainAin has the same value for any i, and is the definition of det A, the determinant of A. This particular expression is the evaluation of det A by the i-th row.(ii) On the other hand, if ij, ai1Aj1+ai2Aj1+⋯+ainAin=0.Results for columns, corresponding to the results (i) and (ii) for rows, also hold.

(i) The expression ai1Ai1+ai2+⋯+ ainAin has the same value for any i, and is the definition of det A, the determinant of A. This particular expression is the evaluation of det A by the i-th row.

(ii) On the other hand, if ij, ai1Aj1+ai2Aj1+⋯+ainAin=0.

Subjects: Mathematics.


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