1.

If A and B are symmetric matrices of the same order, then (AB' - BA') is:1. Skew symmetric matrix2. Symmetric matrix3. Null matrix4. Identity matrix

Answer» Correct Answer - Option 1 : Skew symmetric matrix

Concept:

We have various matrix identities as,

  • For symmetric matrices, A = A' and B = B'
  • For skew-symmetric matrices, A = - A'
  • (A ± B)' = A' ± B'
  • (AB)' = B'A'

 

Calculation:

A and B are symmetric matrices

As we know that for symmetric matrices, we have A = A' and B = B'

(AB' - BA') = AB - BA

∵ (A ± B)' = A' ± B'

(AB - BA)' = (AB)' - (BA)'

∵ (AB)' = B'A'

⇒ (AB - BA)' = B'A' - A'B'

∵ A = A' and B = B

⇒ (AB - BA)' = BA - AB

⇒ (AB - BA)' = - (AB - BA)

Since for skew-symmetric matrices, A = - A'

Hence (AB' - BA') is Skew symmetric matrix.


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