Mathematics - Matrices Question with Solution | TestHub

$A^{\text{'}}=A^T$

$A{A}^{\text{'}}=A^{\text{'}}A$

$B=A^{-1}{A}^{\text{'}}$

$B^{\text{'}}={\left(A^{-1}{A}^{\text{'}}\right)}^{\text{'}}={\left(A^{\text{'}}\right)}^{\text{'}}·{\left(A^{-1}\right)}^{\text{'}}=A{\left(A^{-1}\right)}^{\text{'}}$

$B{B}^{\text{'}}=A^{-1}{A}^{\text{'}}·A{\left(A^{-1}\right)}^{\text{'}}$

       $=A^{-1}·A·A^{\text{'}}·{\left(A^{-1}\right)}^{\text{'}}$     $\left[\because A{A}^{\text{'}}=A^{\text{'}}A\right]$

       $=I·A^{\text{'}}·{\left(A^{-1}\right)}^{\text{'}}$

       $=I·{\left(A^{-1}A\right)}^{\text{'}}$           $\left[\because {\left(CD\right)}^{\text{'}}=D^{\text{'}}{C}^{\text{'}}\right]$

       $=I·I^{\text{'}}$

       $=I^2$                                      $\left[\because I=I^{\text{'}}\right]$

       $=I$