Question

Assertion :The inverse of the matrix $A=[a_{ij}{]}_{n\times n}$, where $a_{ij}=0,i\ge j$ is $B=[a_{ij}^{-1}{]}_{n\times n}$ Reason: The inverse of singular matrix does not exist.

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Assertion is incorrect but Reason is correct

Answer 1

The correct option is D Assertion is incorrect but Reason is correct

$A={\left[a_{ij}\right]}_{n\times n}{a}_{ij}=0i\ge j$

$\Rightarrow \left[\begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ a_{21}=0& a_{22}& a_{23}\\ a_{31}=0& a_3=0& a_{33}\end{array}\right]$

$=\left[\begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ 0& a_{22}& a_{23}\\ 0& 0& a_{33}\end{array}\right]$

$=a_{11}{a}_{22}{a}_{23}$

$5=A^{-1}{A}^{-1}=?$

Minor$=\left[\begin{array}{ccc}{a}_{22}{a}_{33}& 0& 0\\ a_{12}{a}_{33}& a_{11}{a}_{33}& 0\\ a_{12}{a}_{23}-a_{22}{a}_{13}& a_{11}{a}_{23}& a_{11}{a}_{22}\end{array}\right]\left[\begin{array}{ccc}+& -& +\\ -& +& -\\ +& -& +\end{array}\right]$

$A^{-1}=\frac{AdjA}{\left|A\right|}=\frac{1}{a_{11}{a_{22}a}_{33}}\left[\begin{array}{ccc}{a}_{22}{a}_{33}& 0& 0\\ -a_{11}{a}_{33}& a_{11}{a}_{33}& 0\\ a_{12}{a}_{23}-a_{22}{a}_{13}& -a_{11}{a}_{23}& a_{11}{a}_{22}\end{array}\right]$

Option D