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NDA 2022
Exam
NDA
Year
2022
Subject
Mathematics
Chapter
Matrices

NDA 2022 · Mathematics · Matrices

Consider the following statements in respect of square matrices A...

Consider the following statements in respect of square matrices AA and BB of same order: 1. If ABA B is a null matrix, then at last one of AA and BB is a null matrix. 2. If ABA B is an identity matrix, then BA=ABB A=A B. Which of the above statements is/are correct?

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

Let \quad A=\left[1111\begin{array}{ll}1 & 1 \\ 1 & 1\end{array}\right] and B=\left[1−1−11\begin{array}{cc}1 & -1 \\ -1 & 1\end{array}\right]A B=\left[1111\begin{array}{ll}1 & 1 \\ 1 & 1\end{array}\right]\left[1−1−11\begin{array}{cc}1 & -1 \\ -1 & 1\end{array}\right]=\left[0000\begin{array}{ll}0 & 0 \\ 0 & 0\end{array}\right] But A≠0A \neq 0 and B≠0B \neq 0 So, statement I is not correct ∵AB=\because \quad A B=\mathrm{I}⇒B=A−1∴BA=I\Rightarrow B=A^{-1}\therefore \quad B A=I So, statement 2 is correct.
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