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Question
If A & B are two orthogonal matrices of order ′n′, then which of the following is true -
Options
A.AB is orthogonal
B.(AB)2 is orthogonal
C.The value of ||AB|A| is 1 only
D.The value of ||AB|B| is -1 only
Solution
AA′ = I, BB′ = I
(AB) (AB)′ = ABB′A′ = I
[(AB) (AB)] [(AB) (AB)]′ = AB AB(AB)′(AB)′
= ABABB′A′B′A′ = I
||AB| A| = |±A| = (±1)n|A| = 1 or - 1
also ||AB|B| = 1 or - 1
(AB) (AB)′ = ABB′A′ = I
[(AB) (AB)] [(AB) (AB)]′ = AB AB(AB)′(AB)′
= ABABB′A′B′A′ = I
||AB| A| = |±A| = (±1)n|A| = 1 or - 1
also ||AB|B| = 1 or - 1
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