Let A = [a_ij] be a square matrix of order 3 × 3 and C_ij denote cofactor of a_ij in A. If |A| = 5, write the value of a₃₁ C₃₁ + a₃₂ C₃₂ a₃₃ C₃₃.

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Solution

$If A = [a_{i j}] is a square matrix of order n and C_{i j} is a cofactor of a_{i j}, then \sum_{i = 1}^{n} a_{i j} C_{i j} = |A| and \sum_{j = 1}^{n} a_{i j} C_{i j} = |A| Given : |A| = 5 and matrix A is of order 3 \times 3 Since a_{13} C_{13} + a_{23} C_{23} + a_{33} C_{33} represent expansion of A along third column, we get a_{13} C_{13} + a_{23} C_{23} + a_{33} C_{33} = |A| = 5 \Rightarrow a_{13} C_{13} + a_{23} C_{23} + a_{33} C_{33} = 5$

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