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Conjugate of a Matrix

Let A=[ai j] ...

Question

Let A = [a_ij] be a 3 × 3 matrix such that |A| = 5. If C_ij = Cofactor of a_ij in A. Then a₁₁ C₁₁ + a₁₂ C₁₂ + a₁₃ C₁₃ = ________.

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Solution

Given:
|A| = 5

As we know,
Sum of products of elements of row (or column) with their corresponding cofactors = Value of the determinant
and
Sum of products of elements of row (or column) with the cofactors of any other row (or column) = 0

Thus, a₁₁ C₁₁ + a₁₂ C₁₂ + a₁₃ C₁₃ = |A| = 5

Hence, a₁₁ C₁₁ + a₁₂ C₁₂ + a₁₃ C₁₃ = 5.

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