Let $A = [a_{ij}]$ be a square matrix of order n such that $a_{ij}=\left\{\begin{matrix}0,&if\,i≠j\\i,&if\,i=j\end{matrix}\right.$
Statement-1: The inverse of A is the matrix $B = [b_{ij}]$ such that $b_{ij}=\left\{\begin{matrix}0,&if\,i≠j\\\frac{1}{i},&if\,i=j\end{matrix}\right.$
Statement-2: The inverse of a diagonal matrix is a scalar matrix.
Answer & explanation
Correct answer: option 3
We know that the inverse of a diagonal matrix
$D= diag (d_1, d_2, d_3...,d_n)$ is a diagonal matrix given by
$D^{-1}=diag ({d_1}^{-1}, {d_2}^{-1}, {d_3}^{-1},..., {d_n}^{-1})$
$∴B = [b_{ij}]$ is given by
$b_{ij}=\left\{\begin{matrix}0,&if\,i≠j\\\frac{1}{i},&if\,i=j\end{matrix}\right.$
Hence, statement-1 is true and statement-2 is false.