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. none of these

Answer 1

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

$f\left(x\right)=\frac{1+x}{1-x}=\frac{(1+x)(1+x+x^2)}{(1-x)(1+x+x^2)}=\frac{1+2x+2x+x^3}{1-x^3}$
$\therefore f\left(A\right)=\frac{1+2A+2A^2+A^3}{1-A^3}=1+2A+2A^2$, since $A^3=O$