Question

Let $a$ and $b$ be two real numbers such that $a>1,b>1$. If $A=\left[\begin{array}{cc}a& 0\\ 0& b\end{array}\right]$, then $\underset{n\to \infty }{lim}{A}^{-n}$ is

• A. Unit matrix
• B. Null matrix
• C. $2I$
• D. none of these

Answer 1

The correct option is B Null matrix

Given, $A=\left[\begin{array}{cc}a& 0\\ 0& b\end{array}\right]$
$A^2=\left[\begin{array}{cc}{a}^2& 0\\ 0& b^2\end{array}\right]$
$\Rightarrow A^n=\left[\begin{array}{cc}{a}^n& 0\\ 0& b^n\end{array}\right]$
$\Rightarrow A^{-n}=\left[\begin{array}{cc}{a}^{-n}& 0\\ 0& b^{-n}\end{array}\right]$
Now,$\underset{n\to \infty }{lim}{A}^{-n}$
$=\underset{n\to \infty }{lim}\left[\begin{array}{cc}{a}^{-n}& 0\\ 0& b^{-n}\end{array}\right]$
$=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]$
Which is a null matrix.