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NDA 2022
Exam
NDA
Year
2022
Subject
Mathematics
Chapter
Matrices

NDA 2022 · Mathematics · Matrices

Let A and B be non-singular matrices of the same order such that A...

Let AA and BB be non-singular matrices of the same order such that AB=AA B=A and BA=BB A=B. Which of the following statements is/are correct? 1. A2=A2.A^2=A 2.A B^2=A^2 BSelectthecorrectanswerusingthecodegivenbelow:Select the correct answer using the code given below :

Options

  1. A1 only
  2. B2 only
  3. CBoth 1 and 2
  4. DNeither 1 nor 2

Correct answer

B. 2 only

Step-by-step solution

 Given that ∣A∣≠0,∣B∣≠0AB=A⇒A−1AB=A−1A⇒B=Iand BA=B⇒B−1BA=B−1BA=I∴A2=I2=I=A\begin{aligned} & \text{ Given that }|A| \neq 0,|B| \neq 0 \\ & A B=A \Rightarrow A^{-1} A B=A^{-1} A \Rightarrow B=I \\ & \text{and } B A=B \Rightarrow B^{-1} B A=B^{-1} B \\ & A=I \\ & \therefore \quad A^2=I^2=I=A\end{aligned} So, statement 1 is correct \begin{aligned} & \text{ Now, } \begin{aligned} & A B^2=I^3=I \\ & A^2 B=I^3=I \\ & \therefore \quad A B^2=A^2 B\end{aligned}\end{aligned}$ So, statement 2 is correct
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