Haryana Police Constable Model Paper 2

Concept:
Let A be any square matrix.
$\left(\mathrm{adj}{\mathrm{A}}^{\mathrm{T}}\right)=|\mathrm{A}{|}^{\mathrm{T}}({\mathrm{A}}^{\mathrm{T}{)}^{-1}}$
${\left(\mathrm{adj}\mathrm{A}\right)}^{\mathrm{T}}$=${\left(\left|\mathrm{A}\right|{\mathrm{A}}^{-1}\right)}^{\mathrm{T}}$
$\left(\mathrm{adj}{\mathrm{A}}^{\mathrm{T}}\right)={\left(\mathrm{adj}\mathrm{A}\right)}^{\mathrm{T}}$
Calculations:
Given, A is a square matrix.
Consider, $\left(\mathrm{adj}{\mathrm{A}}^{\mathrm{T}}\right)-{\left(\mathrm{adj}\mathrm{A}\right)}^{\mathrm{T}}$
$=|{\mathrm{A}}^{\mathrm{T}}|{\left({\mathrm{A}}^{\mathrm{T}}\right)}^{-1}-{\left(\left|\mathrm{A}\right|{\mathrm{A}}^{-1}\right)}^{\mathrm{T}}$
$=|\mathrm{A}{|}^{\mathrm{T}}{\left({\mathrm{A}}^{\mathrm{T}}\right)}^{-1}-({\left|\mathrm{A}\right|}^{\mathrm{T}}{\left({\mathrm{A}}^{-1}\right)}^{\mathrm{T}}$
$=|\mathrm{A}{|}^{\mathrm{T}}{\left({\mathrm{A}}^{-1}\right)}^{\mathrm{T}}-({\left|\mathrm{A}\right|}^{\mathrm{T}}{\left({\mathrm{A}}^{-1}\right)}^{\mathrm{T}}$
= 0
= Null Matrix
Hence, If A is a square matrix, then $\mathrm{adj}{\mathrm{A}}^{\mathrm{T}}-{\left(\mathrm{adj}\mathrm{A}\right)}^{\mathrm{T}}$ equal to null matrix.