DeterminantHard
Question
Matrix A is such that A2 = 2A - I, where I is the identity matrix. Then for n ≥ 2, An =
Options
A.nA - (n - 1)I
B.nA - I
C.2n - 1 A - (n - 1)I
D.2n - 1 A - I
Solution
A2 - 2A + I = 0 ⇒ (A - I)2 = 0
An = (A - I + I)n
= nC0 (A - I)n + ....... + nCn-2 (A - I)2 . In - 2 + nCn-1(A - I) . In - 1 + nCn In
= 0 + 0 + .......... + 0 + n(A – I) + I
= nA - (n - 1)I
An = (A - I + I)n
= nC0 (A - I)n + ....... + nCn-2 (A - I)2 . In - 2 + nCn-1(A - I) . In - 1 + nCn In
= 0 + 0 + .......... + 0 + n(A – I) + I
= nA - (n - 1)I
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