Question

Let $f\left(x\right)=\frac{1+x}{1-x}.$ If $A$ is matrix for which $A^3=O,$ then $f\left(A\right)$ is

• A. $I+A+A^2$
• B. $I+2A+2A^2$
• C. $I-A-A^2$
• D. $O$

Answer 1

The correct option is B $I+2A+2A^2$

Given: $A^3=O,f\left(x\right)=\frac{1+x}{1-x}$
Replacing $x$ with $A$ in the given expression and on cross multiplying, we have: $(I-A)f\left(A\right)=I+A\Rightarrow f\left(A\right)=(I-A{)}^{-1}(I+A)$
We know, $(I-A{)}^{-1}=I+A+A^2+A^3+...$
But we have $A^3=O$
$\Rightarrow f\left(A\right)=(I+A+A^2)(1+A)=I+2A+2A^2$