Question

show that all the positive integers integral power of symmetric matrix are symmetric

Answer 1

uppose AA is the square matrix of order say mm,then Observe that (An)t=(At)n(An)t=(At)n ,

so for your first case for Symmetric matrices At=AAt=A , so An=A.A.A.....AAn=A.A.A.....A , nn times =At.At.At....At=At.At.At....At , nn times =(At)n=(An)t=(At)n=(An)t.so AnAn is symmetric for any positive integer nn.

For the second case also since At=−AAt=−A,so (At)n=−A.−A.−A....−A(At)n=−A.−A.−A....−A nn times,=(−1)n.(A)n=(−1)n.(A)nso if nn is odd it will be skew-symmetric and for nn even it will be symmetric.Hope this helps!

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Answer 2

Answer:ssee attachment

Step-by-step explanation: