Explanation

(a)
$A = diag\ (d_{1},\ d_{2},\ d_{3},\ \cdots,\ d_{n})$
$\Rightarrow |A| = \left( d_{1} \times d_{2} \times d_{3} \times d_{4} \times \cdots \times d_{n} \right)$
Now,
Cofactor of $d_{1}$is$d_{2}d_{3}\ \cdots d_{n}$
Cofactor of $d_{2}$ is $d_{1} \times d_{3} \times d_{4}\ \cdots d_{n}$
Cofactor of $d_{3}$ is $d_{1} \times d_{2} \times d_{4}\ \cdots d_{n}$
$\vdots$
Cofactor of $d_{n}$ is $d_{1} \times d_{2} \times d_{3}\ \cdots d_{n - 1}$
$\Rightarrow A^{- 1} = \frac{1}{|A|}adj(A) = diag(d_{1}^{- 1},\ d_{2}^{- 1},\ d_{3}^{- 1},\cdots,d_{n}^{- 1})$
Hence, $A^{- 1}$ is also a diagonal matrix.