Q194 / 270 · NDA 2025 (Phase 2) · Mathematics

Consider the following statements:
Statement-I: If $X$ is an $n \times n$ matrix, then $\det(mX) = m^n \det(X)$, where $m$ is a scalar.
Statement-II: If $Y$ is a matrix obtained from $X$ by multiplying any row or column by a scalar $m$, then $\det(Y) = m\det(X)$.
Which one of the following is correct in respect of the above statements?

  • Correct answer
    A
    Both Statement-I and Statement-II are correct and Statement-II explains Statement-I
  • B
    Both Statement-I and Statement-II are correct but Statement-II does not explain Statement-I
  • C
    Statement-I is correct but Statement-II is not correct
  • D
    Statement-I is not correct but Statement-II is correct

Solution

Statement-I states that for an $n \times n$ matrix $X$, $\det(mX) = m^n \det(X)$. This is a standard property of determinants and is correct.

Statement-II states that if a single row or column of a matrix is multiplied by a scalar $m$, the determinant of the resulting matrix is $m$ times the determinant of the original matrix. This is also a fundamental property of determinants and is correct.

Multiplying a matrix $X$ by a scalar $m$ is equivalent to multiplying each of its $n$ rows by $m$. By applying the property given in Statement-II $n$ times (once for each row), we can factor out $m$ from each of the $n$ rows, yielding $m^n \det(X)$.

Thus, Statement-II provides the logical reasoning required to prove Statement-I. Both statements are correct, and Statement-II explains Statement-I.

Hence, the correct answer is option: 'A'.

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