Mathematics - Matrix Question with Solution | TestHub

Given,

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

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

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

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

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

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

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

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

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

$=I·I=I$