Manipal MET2015MathematicsMatrices

If A is a square matrix such that A^2=A and (1+A)^n=I+ A , then is equal to

Options

A2 n-1
B2^n-1
C2 n+1
DNone of these

Correct answer

B. 2^n-1

Step-by-step solution

We have, A^2=A (I+A)^2=(I+A)(I+A)=I+2 A+A^2=I+3 A and (I+A)^3=(I+A)^2(I+A) array lr =(I+3 A)(I+A) & [ (I+A)^2=I+3 A ] =I+4 A+3 A^2 & =I+7 A & [ A^2=A ] array Thus, we have (I+A)^2=I+3 A and (I+A)^3=I+7 A (I+A)^2=I+ (2^2-1 ) A and (I+A)^3=I+ (2^3-1 ) A Hence, (I+A)^n=I+ (2^n-1 ) A aligned & & (I+A)^n & =I+ A & & & =2^n-1 aligned

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