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NDA 2 2025 Maths, question 44

Topic: Matrices and determinants

Q44

Consider the following statements :

Statement-I : If XX is an n×nn\times n matrix, then det(mX)=mndet(X)\text{det}(mX)=m^{n}\text{det}(X), where mm is a scalar.

Statement-II : If YY is a matrix obtained from XX by multiplying any row or column by a scalar mm, then det(Y)=mdet(X)\text{det}(Y)=m\text{det}(X).

Which one of the following is correct in respect of the above statements?

  1. (a) Both Statement-I and Statement-II are correct and Statement-II explains Statement-ICorrect answer
  2. (b) Both Statement-I and Statement-II are correct but Statement-II does not explain Statement-I
  3. (c) Statement-I is correct but Statement-II is not correct
  4. (d) Statement-I is not correct but Statement-II is correct

Solution.

Answer: (a) Both Statement-I and Statement-II are correct and Statement-II explains Statement-I

  1. Multiplying one row by m multiplies the determinant by m.
  2. In mX every one of the n rows is multiplied by m, so applying that rule n times gives mⁿdet(X).
  3. Thus both statements are true and II explains I.