Question

Statement 1 : If $A=\left[\begin{array}{ccc}a& 0& 0\\ 0& b& 0\\ 0& 0& c\end{array}\right]$ then $A^{-1}=\left[\begin{array}{ccc}\frac{1}{a}& 0& 0\\ 0& \frac{1}{b}& 0\\ 0& 0& \frac{1}{c}\end{array}\right]$
Statement 2 : The inverse of a diagonal matrix is a diagonal matrix.

• A. Statement -1 is false, Statement-2 is true
• B. Statement -1 is true, Statement-2 is true;
  Statement -2 is a correct explanation for Statement-1
• C. Statement -1 is true, Statement-2 is true;
  Statement -2 is not a correct explanation for Statement-1
• D. Statement -1 is true, Statement-2 is false

Answer 1

Answer: (B)

$A^{-1}=\frac{1}{\det A}\text{adj}A$
$=\frac{1}{abc}\left[\begin{array}{ccc}bc& 0& 0\\ 0& ca& 0\\ 0& 0& ab\end{array}\right]$
$=\left[\begin{array}{ccc}\frac{1}{a}& 0& 0\\ 0& \frac{1}{b}& 0\\ 0& 0& \frac{1}{c}\end{array}\right]$
The inverse of a diagonal matrix is a diagonal matrix. Both true but Statement 2 is not correct reason of Statement 1.