UPSC NDA 2 2022 Mathematics Solved Paper

Concept:The sum of the products of the elements of any row with their respective cofactors equals the determinant of the matrix.Explanation:For a $3 \times 3$ matrix $A = \begin{pmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{pmatrix}$, the determinant $\Delta$ is given by expanding along the first row.The cofactors $C_{11}, C_{12}, C_{13}$ correspond to $a_{11}, a_{12}, a_{13}$ respectively.By definition, $\Delta = a_{11}C_{11} + a_{12}C_{12} + a_{13}C_{13}$.Thus the given expression $a_{11}C_{11} + a_{12}C_{12} + a_{13}C_{13}$ is exactly equal to $\Delta$.This is a standard property of determinants used for expansion.Answer:$\Delta$ (Option C).