Question

# Matrix $$\mathrm{A}$$ is such that $$\mathrm{A}^{2}=2\mathrm{A}-1$$ where 1 is the unit matrix Then for $$\mathrm{n}\geq 2,\ \mathrm{A}^{\mathrm{n}}=$$

A
nA(n1)I
B
nAI
C
2n+1A(n1)I
D
2n+1A1

Solution

## The correct option is B $$nA-(n-1)I$$$$A^2=2A-1$$$$A^n=A^{n-2}A^2$$$$=A^{n-2}(2A-1)$$$$=2 A^{n-1}-A^{n-2}$$$$A^2-2A+I=0$$$$(A-I)=0$$$$A=I$$$$A^n=I$$Put in all options only (A) comes out IMaths

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