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NDA 2024
Exam
NDA
Year
2024
Subject
Mathematics
Chapter
Matrices

NDA 2024 · Mathematics · Matrices

Consider the following statements in respect of two non-singular...

Consider the following statements in respect of two non-singular matrices A and B of the same order nn : 1. adj⁡(AB)=(adj⁡A)(adjB⁡)\operatorname{adj}(\mathrm{AB})=(\operatorname{adj} \mathrm{A})(\operatorname{adjB}) 2. adj⁡(AB)=adj⁡(BA)\operatorname{adj}(\mathrm{AB})=\operatorname{adj}(\mathrm{BA}) 3. (AB)adj⁡(AB)−∣AB∣(\mathrm{AB}) \operatorname{adj}(\mathrm{AB})-|\mathrm{AB}|\mathrm{I}n_n is a null matrix of order nn How many of the above statements are correct?

Options

  1. ANone
  2. BOnly one statement
  3. COnly two statements
  4. DAll three statements

Correct answer

B. Only one statement

Step-by-step solution

Statements: 1. adj⁡(AB)=(adj⁡A)(adjB⁡)\operatorname{adj}(\mathrm{AB})=(\operatorname{adj} \mathrm{A})(\operatorname{adjB}) Statement 1 is not correct. 2. AB≠BA∴adj⁡(AB)≠adj⁡(BA)\mathrm{AB} \neq \mathrm{BA}\therefore \operatorname{adj}(\mathrm{AB}) \neq \operatorname{adj}(\mathrm{BA}) Statement 2 is not correct. 3. (AB)adj⁡(AB)−∣AB∣In\begin{aligned} & (\mathrm{AB}) \operatorname{adj}(\mathrm{AB})-|\mathrm{AB}| \mathrm{I}_n \end{aligned}⇒∣AB∣In−∣AB∣In\begin{aligned} \Rightarrow|\mathrm{AB}| \mathrm{I}_n-|\mathrm{AB}| \mathrm{I}_n \end{aligned}[\operatorname{Aadj} \mathrm{A}=|\mathrm{A}| I\mathrm{I}]=$ Null matrix Hence, statement 3 is correct.
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