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Let A, B, C, D be (not necessarily ) r...

Let ` A, B, C, D ` be (not necessarily ) real matrices such that `A^(T) = BCD , B^(T) =CDA, C^(T) = DAB and D^(T) =ABC` for the matrix `S = ABCD` the least value of k such that `S^(k) = S` is

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The correct Answer is:
3
`S = ABCD = A ( BCD) = A A^(T)` …(i) `therefore S^(3) = (ABCD) (ABCD) (ABCD)`
`= (ABC) (DAB) (CDA) (BCD)`
`= D^(T) C^(T) B^(T) A^(T) = (BCD)^(T) A^(T)`
`= (A^(T))^(T) A^(T) = A A^(T) = S`
`rArr S^(3) = S`
Hence, least valuse of k is 3.
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