Question

If the matrix $A=\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]$, then $I+A+A^2+...........$ upto $A^{\infty }=$.......

• A. $\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$
• B. $\left[\begin{array}{cc}\frac{1}{2}& \frac{1}{3}\\ \frac{1}{-2}& 0\end{array}\right]$
• C. $\left[\begin{array}{cc}\frac{-1}{2}& \frac{1}{-3}\\ \frac{1}{-2}& 0\end{array}\right]$
• D. $\left[\begin{array}{cc}\frac{1}{2}& \frac{1}{-3}\\ \frac{1}{-2}& 0\end{array}\right]$

Answer 1

Answer: (D)

$\left[\begin{array}{cc}\frac{1}{2}& \frac{1}{-3}\\ \frac{1}{-2}& 0\end{array}\right]$

We have
$1+A+A^2+.......+\infty =I(I-A{)}^{-1}$
$=(I-A{)}^{-1}$

$={\left[\begin{array}{cc}0& -2\\ -3& -3\end{array}\right]}^{-1}$

$=\frac{1}{-6}\left[\begin{array}{cc}-3& 2\\ 3& 0\end{array}\right]$

$=\left[\begin{array}{cc}\frac{1}{2}& \frac{1}{-3}\\ \frac{1}{-2}& 0\end{array}\right]$