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NDA 2 2025 Maths, question 45

Topic: Matrices and determinants

Q45

Consider the following statements about the matrix

M=[712348572829651748]M=\begin{bmatrix} 71 & 23 & 48 \\ 57 & 28 & 29 \\ 65 & 17 & 48 \end{bmatrix} :

Statement-I : The inverse of MM does not exist.

Statement-II : MM is non-singular.

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-I
  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 correctCorrect answer
  4. (d) Statement-I is not correct but Statement-II is correct

Solution.

Answer: (c) Statement-I is correct but Statement-II is not correct

  1. The first column equals the sum of the second and third: 71=23+48, 57=28+29 and 65=17+48.
  2. The columns are dependent, so det(M)=0.
  3. Its inverse does not exist, and it is singular; I is true and II false.