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NDA 2023
Exam
NDA
Year
2023
Subject
Mathematics
Chapter
Matrices

NDA 2023 · Mathematics · Matrices

Let A and B be symmetric matrices of same order, then which one of...

Let AA and BB be symmetric matrices of same order, then which one of the following is correct regarding (AB−BA)(A B-B A) ? 1. Its diagonal entries are equal but non-zero 2. The sum of its non-diagonal entries is zero 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

For symmetric matrices, A′=A,B′=B(AB−BA)′=BA−AB=−(AB−BA)⇒(AB−BA)A^{\prime}=A, B^{\prime}=B(A B-B A)^{\prime}=B A-A B=-(A B-B A)\Rightarrow(A B-B A) is a skew symmetric matrix ⇒(AB−BA)\Rightarrow(A B-B A) matrix have all diagonal element as zero (statement 1 is incorrect) Sum of all non diagonal elements in (AB−BA)=0∴(A B-B A)=0\therefore Only 2 is correct.
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