If A= pmatrix2&3\0&-1 pmatrix then the value of (A^4)+ (A^10-( Adj(2A))^10 ) is equal to

Question

Accepted Answer

Correct answer: 16. (16) 2 Aadj ⁡ (2A )=12 AlI ⇒ Aadj ⁡ (2A )=-4I … … (i ) Now, E= (A^4 )+(A^10- ( adj ⁡ (2A ) )^10) = (-2 )^4+ (A^20-A^10 ( adj ⁡2A )^10) |A |^10 =16+ (A^20- ( Aadj ⁡ (2A ) )^10) |A |^10 =16+ (A^20-2^10I)2^10 (from) Now, characteristic roots of A are 2 and -1 . So, characteristic roots of A^20 are 2^10and 1 . Hence, (A^20-2^10I)(A^20-I)=0 ⇒ ∣A^20-2^10 Il=0 ( asA^20 I ) ⇒E=16

$If A = (20 3 - 1) then the value of det (A^{4}) + det (A^{10} - (Adj (2 A))^{10}) is equal to$

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