Question

Let $A=\left[a_{ij}\right]$ be a square matrix of order $n$ such that
$a_{ij}=\{\begin{array}{c}0ifi\ne j\\ iifi=j\end{array}$
Statement - 1 : The inverse of A is matrix $B=\left[b_{ij}\right]$ such that
$b_{ij}=\{\begin{array}{c}0ifi\ne j\\ \frac{1}{i}ifi=j\end{array}$
Statement - 2 : The inverse of a diagonal matrix is a scalar matrix.

• A. Statement-1 is True, Statement-2 is True, Statement-2 is a correct explanation for Statement-1
• B. Statement-1 is True, Statement-2 is True, Statement-2 is NOT a correct explanation for Statement-1
• C. Statement-1 is True, Statement-2 is False.
• D. Statement-1 is False, Statement-2 is True.

Answer 1

Answer: (C)

Statement-1 is True, Statement-2 is False.

let order of the matrix $n$ is $3$
$\therefore A=\left[\begin{array}{ccc}i& 0& 0\\ 0& i& 0\\ 0& 0& i\end{array}\right]$
$|A|=i^3$
$adjA=\left[\begin{array}{ccc}{i}^2& 0& 0\\ 0& i^2& 0\\ 0& 0& i^2\end{array}\right]$
$\Rightarrow B=A^{-1}=\frac{adjA}{|A|}=\left[\begin{array}{ccc}\frac{1}{i}& 0& 0\\ 0& \frac{1}{i}& 0\\ 0& 0& \frac{1}{i}\end{array}\right]$
The inverse of a diagonal matrix is also a diagonal matrix. It need not be a scalar matrix
Hence, Statement-1 is True, Statement-2 is False.