Question

In question 18, write the value of a₁₁ C₂₁ + a₁₂ C₂₂ + a₁₃ C₂₃.

Answer 1

We know that in a square matrix of order n, the sum of the products of elements of a row (or a column) with the cofactors of the corresponding elements of some other row (or column ) is zero. Therefore,
\[A = \left[ a_{i j} \right]\text{ is a square matrix of order n }. \]
\[ \Rightarrow \sum^n_{j = 1} a_{i j} C_{k j} = 0\text{ and } \sum^n_{i = 1} a_{i j} C_{i k} = 0\] 
\[ \Rightarrow a_{11} C_{21} + a_{12} C_{22} + a_{13} C_{23} = 0 \left[\text{ Since the elements are of first row and the cofactors are of elements of second row }\right]\] 
\[ \Rightarrow a_{11} C_{21} + a_{12} C_{22} + a_{13} C_{23} = 0\]