Explanation

(b)
$A = \begin{pmatrix}
a & 0 \\
0 & b
\end{pmatrix}$
$\Rightarrow A^{2} = \begin{pmatrix}
a & 0 \\
0 & b
\end{pmatrix}\begin{pmatrix}
a & 0 \\
0 & b
\end{pmatrix} = \begin{pmatrix}
a^{2} & 0 \\
0 & b^{2}
\end{pmatrix}$
$\Rightarrow A^{3} = \begin{pmatrix}
a^{2} & 0 \\
0 & b^{2}
\end{pmatrix}\begin{pmatrix}
a & 0 \\
0 & b
\end{pmatrix} = \begin{pmatrix}
a^{3} & 0 \\
0 & b^{3}
\end{pmatrix}$
$\Rightarrow A^{n} = \begin{pmatrix}
a^{n} & 0 \\
0 & b^{n}
\end{pmatrix}$
$\Rightarrow \left( A^{n} \right)^{- 1} = \frac{1}{a^{n}b^{n}}\begin{pmatrix}
a^{n} & 0 \\
0 & b^{n}
\end{pmatrix} = \begin{pmatrix}
a^{- n} & 0 \\
0 & b^{- n}
\end{pmatrix}$
$\Rightarrow \lim_{n \rightarrow \infty}\left( A^{n} \right)^{- 1} = \begin{pmatrix}
0 & 0 \\
0 & 0
\end{pmatrix}$ as $a > 1\$and$\ b > 1$