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NDA 1 2024 Maths, question 11

Topic: Matrices and determinants

Q11

Consider the following statements in respect of two non-singular matrices AA and BB of the same order nn :

  1. adj(AB)=(adjA)(adjB)\text{adj}(AB)=(\text{adj}A)(\text{adj}B)

  2. adj(AB)=adj(BA)\text{adj}(AB)=\text{adj}(BA)

  3. (AB)adj(AB)−∣AB∣In(AB)\text{adj}(AB)-|AB|I_{n} is a null matrix of order nn

How many of the above statements are correct ?

  1. (a) None
  2. (b) Only one statementCorrect answer
  3. (c) Only two statements
  4. (d) All three statements

Solution.

Answer: (b) Only one statement

  1. The general identity reverses order: adj(AB)=(adj B)(adj A), so statement 1 need not hold.
  2. AB and BA need not have equal adjugates either, so statement 2 is not a general identity.
  3. Statement 3 is always true because M adj M=(det M)I.
  4. Thus only one statement is correct.