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If A = [ 0 − x x 0 ] , B = [ 0 1 1 0 ] , and x 2 = − 1 , then show that ( A + B ) 2 = A 2 + B 2 .

0 3 Views | Posted 3 months ago
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    Answered by

    Payal Gupta | Contributor-Level 10

    3 months ago

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

    G i v e n     t h a t     A = [ 0 − x x 0 ]     a n d     B = [ 0 1 1 0 ] L . H . S .                       ( A + B ) 2 = ( A + B ) . ( A + B )                                                     = [ ( 0 − x x 0 ) + ( 0 1 1 0 ) ] . [ ( 0 − x x 0 ) + ( 0 1 1 0 ) ]                                                       = [ 0 + 0 − x + 1 x + 1 0 + 0 ] . [ 0 + 0 − x + 1 x + 1 0 + 0 ]                                                       = [ 0 + ( − x + 1 ) ( x + 1 ) 0 + 0 0 + 0 ( x + 1 ) ( − x + 1 ) + 0 ]                                                         = [ 1 − x 2 0 0 1 − x 2 ] P u t     x 2 = − 1                                                   = [ 1 + 1 0 0 1 + 1 ] = [ 2 0 0 2 ] R . H . S .         A 2 + B 2 = A . A + B . B                                             = [ 0 − x x 0 ] . [ 0 − x x 0 ] + [ 0 1 1 0 ] . [ 0 1 1 0 ]                                             = [ 0 − x 2 0 + 0 0 + 0 − x 2 + 0 ] + [ 0 + 1 0 + 0 0 + 0 1 + 0 ]                                               = [ − x 2 0 0 − x 2 ] + [ 1 0 0 1 ] = [ − x 2 + 1 0 + 0 0 + 0 − x 2 + 1 ]                                                 = [ − x 2 + 1 0 0 − x 2 + 1 ] = [ 1 + 1 0 0 1 + 1 ]         [ ?     x 2 = − 1 ]                                                   = [ 2 0 0 2 ] H e n c e ,                             L . H . S . = R . H . S .                                                           ( A + B ) 2 = A 2 + B 2

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