If $ A $ is a non-null diagonal matrix of order $ 3 $ such that $ A^{4}=A^{2} $, then the possible number of matrices $ A $ are

Question

If $ A $ is a non-null diagonal matrix of order $ 3 $ such that $ A^{4}=A^{2} $, then the possible number of matrices $ A $ are, $ 27 $, $ 26 $, $ 8 $, $ 7 $

MARKS App · Accepted Answer

Let, A = d 1 0 0 0 d 2 0 0 0 d 3 A 2 = d 1 2 0 0 0 d 2 2 0 0 0 d 3 2 & A 4 = d 1 4 0 0 0 d 2 4 0 0 0 d 3 4 ∵ A 4 = A 2 ⇒ d 1 4 = d 1 2 ⇒ d 1 = 0,1 , - 1 d 2 4 = d 2 2 ⇒ d 2 = 0,1 , - 1 d 3 4 = d 3 2 ⇒ d 3 = 0,1 , - 1 ∵ A can't be a null matrix so the total number of possible matrices is = 3 3 - 1 = 26

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