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i want answer in this sheet of question 1 and 3 please describe the answer thoroughly

rock , 3 Years ago
Grade 12
anser 1 Answers
raj kumar

Last Activity: 3 Years ago

1st question-answer:
here, symmetric matrices mean AT=A  or BT=B
so,   given question X= AB+BA   -----> (1)
let xT= (AB+BA)T     [since,t=transpose]
         = (AB)T+(BA)T       [ if (AB)T then (AB)T=BTAT]
          =BTAT+ATBT          [we know that symmetric matrix formula AT=A or BT=B]
           = BA+AB
           = x     [ from (1) ]
therefore, XT=X ---------->(2)
given the question, Y  = AB-BA ------->(3)
                    we can write as Y = – (BA-AB)  =>  -Y = (BA-AB) --------->(4)
let YT=(AB-BA)T
          = (AB)T-(BA)T    [SINCE (AB)T=BTAT]
            = BTAT-ATBT      [symmetric formula BT=B]
          =BA-AB
          =    -Y
therefore, YT=  -Y  --------->(5)
similarly, given question (XY)T (or)  (XY)I
  (XY)T=YTXT      [FROM (2) & (5)]
           =   -YX
 Therefore, (XY)T​ -YX.

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