Question

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.

Answer 1

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