UPSC NDA 2 2022 Mathematics Solved Paper

Concept:For any square matrix, the sum of the products of elements from one row with the cofactors from a different row is always zero.Explanation:Here $C_{11}$, $C_{12}$, $C_{13}$ are the cofactors of $a_{11}$, $a_{12}$, $a_{13}$ respectively.We are evaluating $a_{21}C_{11} + a_{22}C_{12} + a_{23}C_{13}$.Write the cofactors explicitly:$C_{11} = a_{22}a_{33} - a_{23}a_{32}$$C_{12} = a_{23}a_{31} - a_{21}a_{33}$$C_{13} = a_{21}a_{32} - a_{22}a_{31}$Substitute into the expression:$a_{21}C_{11} + a_{22}C_{12} + a_{23}C_{13} = a_{21}(a_{22}a_{33} - a_{23}a_{32}) + a_{22}(a_{23}a_{31} - a_{21}a_{33}) + a_{23}(a_{21}a_{32} - a_{22}a_{31})$Expand each term:$= a_{21}a_{22}a_{33} - a_{21}a_{23}a_{32} + a_{22}a_{23}a_{31} - a_{21}a_{22}a_{33} + a_{23}a_{21}a_{32} - a_{22}a_{23}a_{31}$Notice that the first and fourth terms cancel: $a_{21}a_{22}a_{33} - a_{21}a_{22}a_{33} = 0$Similarly, the second and fifth terms: $-a_{21}a_{23}a_{32} + a_{23}a_{21}a_{32} = 0$ (order does not matter)The third and sixth terms: $a_{22}a_{23}a_{31} - a_{22}a_{23}a_{31} = 0$All terms cancel, leaving $0$.Answer:$0$ (Option A).