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Express the matrix [ 2   3 1 1 − 1 2 4 1 2 ] as the sum of a symmetric and a skew-symmetric matrix.

0 5 Views | Posted 3 months ago
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  • 1 Answer

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    Answered by

    Payal Gupta | Contributor-Level 10

    3 months ago

    This is a long answer type question as classified in NCERT Exemplar

    We  know  that any  square  matrix  can  be  expressed  as  the  sum  of  symmetric  and  skew  symmetric  matrix  i.e. A = 1 2 [ A + A ' ] + 1 2 [ A − A ' ] S o                                           P = 1 2 [ ( 2 3 1 1 − 1 2 4 1 2 ) + ( 2 1 4 3 − 1 1 1 2 2 ) ]                                                                 = 1 2 [ 2 + 2 3 + 1 1 + 4 1 + 3 − 1 − 1 2 + 1 4 + 1 1 + 2 2 + 2 ] = 1 2 [ 4 4 5 4 − 2 3 5 3 4 ]                                                                   = [ 2 2 5 2 2 − 1 3 2 5 2 3 2 2 ]                                                       P ' = [ 2 2 5 2 2 − 1 3 2 5 2 3 2 2 ] = P A s     P ' = P                 ∴ P     i s     a     s y m m e t r i c     m a t r i x . N o w ,           Q = 1 2 [ A − A ' ]                                           = 1 2 [ ( 2 3 1 1 − 1 2 4 1 2 ) − ( 2 1 4 3 − 1 1 1 2 2 ) ]                                           = 1 2 [ 2 − 2 3 − 1 1 − 4 1 − 3 − 1 + 1 2 − 1 4 − 1 1 − 2 2 − 2 ] = 1 2 [ 0 2 − 3 − 2 0 1 3 − 1 0 ]                                                                   = [ 0 1 − 3 2 − 1 0 1 2 3 2 − 1 2 0 ] = − [ 0 − 1 3 2 1 0 − 1 2 − 3 2 1 2 0 ] = − Q A s     Q = − Q                 ∴ Q     i s     a     s k e w     s y m m e t r i c     m a t r i x . S o                                   A = P + Q                                             A = [ 2 2 5 2 2 − 1 3 2 5 2 3 2 2 ] + [ 0 1 − 3 2 − 1 0 1 2 3 2 − 1 2 0 ]                                                       = [ 2 + 0 2 + 1 5 2 − 3 2 2 − 1 − 1 + 0 3 2 + 1 2 5 2 + 3 2 3 2 &min

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