UPSC NDA II 2025 Mathematics Paper

Concept:When a scalar multiplies every element of a matrix, the determinant scales by the scalar raised to the matrix order. Multiplying a single row or column scales the determinant by the scalar only once.Explanation:Statement I: For an $n \times n$ matrix $X$, the matrix $mX$ has each row multiplied by $m$. Since the determinant is linear in each row, the factor $m$ appears $n$ times, giving $\det(mX) = m^n \det(X)$. Statement I is correct.Statement II: If $Y$ is obtained from $X$ by multiplying any one row (or column) by $m$, then the determinant picks that factor exactly once: $\det(Y) = m \det(X)$. This is a basic property of determinants. Statement II is also correct.Statement II directly explains Statement I. Multiplying all $n$ rows of $X$ by $m$ is equivalent to applying the rule of Statement II $n$ times (once per row). Therefore, Statement II is the reason for Statement I.Answer:Option A: Both Statement-I and Statement-II are correct and Statement-II explains Statement-I.