A) \[{{2}^{n-1}}A-(n-1)I\]
B) \[{{2}^{n-1}}A-I\]
C) \[nA-(n-1)I\]
D) \[nA-I\]
Correct Answer: C
Solution :
Given, \[{{A}^{2}}=2A-I\] Now, \[{{A}^{3}}={{A}^{2}}.A\] \[=(2A-I)A\] \[=2{{A}^{2}}-A\] \[=2(2A-I)-A\] \[=3A-2I\] Now, \[{{A}^{4}}={{A}^{3}}.A\] \[=(3A-2I).A\] \[=3{{A}^{2}}-2A\] \[=3(2A-I)-2A\] \[=4A-3I\] Similarly, \[{{A}^{n}}=nA-(n-1)I\]You need to login to perform this action.
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