Mathematical ReasoningHard
Question
If A = 
, then which one of the following statements is not correct ?
, then which one of the following statements is not correct ?Options
A.A3 − I = A(A − I)
B.A3 + I = A(A3 − I)
C.A2 + I = A(A2 − I)
D.A4 − I = A(A2 + I)
Solution
A2 = 
= - I
A3 = - A
A4 = - A2 = I
A5 = A
Now
(1) A3 - I = - A - I
A(A - I) = A2 - A = - I - A
(2) A3 + I = - A + I
A(A3 - I) = A(-A - I) = - A2 - A = I - A
(3) A2 + I = - A(I + I) = - 2A
A2 + I ≠ A(A2 - I)
(4) A4 - I = I - I = 0
A2 + I = - I + I = 0
= - IA3 = - A
A4 = - A2 = I
A5 = A
Now
(1) A3 - I = - A - I
A(A - I) = A2 - A = - I - A
(2) A3 + I = - A + I
A(A3 - I) = A(-A - I) = - A2 - A = I - A
(3) A2 + I = - A(I + I) = - 2A
A2 + I ≠ A(A2 - I)
(4) A4 - I = I - I = 0
A2 + I = - I + I = 0
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