Grade 12Algebra
i want answer in this sheet of question 1 and 3 please describe the answer thoroughly
i want answer in this sheet of question 1 and 3 please describe the answer thoroughly
1 Answer
raj kumar
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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