JEE MAIN - Mathematics (2014 (Offline) - No. 7)
If $$A$$ is a $$3 \times 3$$ non-singular matrix such that $$AA'=A'A$$ and
$$B = {A^{ - 1}}A',$$ then $$BB'$$ equals:
$$B = {A^{ - 1}}A',$$ then $$BB'$$ equals:
$${B^{ - 1}}$$
$$\left( {{B^{ - 1}}} \right)'$$
$$I+B$$
$$I$$
Explanation
$$BB' = B\left( {{A^{ - 1}}A'} \right)' = B\left( {A'} \right)'\left( {{A^{ - 1}}} \right)' = BA\left( {{A^{ - 1}}} \right)'$$
$$ = \left( {{A^{ - 1}}A'} \right)\left( {A\left( {{A^{ - 1}}} \right)'} \right)$$
$$ = {A^{ - 1}}A.A'.\left( {{A^{ - 1}}} \right)'\,\,\,\,\,\,$$ $$\left\{ {} \right.$$ as $$\,\,\,\,\,\,$$ $$AA' = A'A$$ $$\left. \, \right\}$$
$$ = I\left( {{A^{ - 1}}A} \right)'$$
$$ = I.I = {I^2} = I$$
$$ = \left( {{A^{ - 1}}A'} \right)\left( {A\left( {{A^{ - 1}}} \right)'} \right)$$
$$ = {A^{ - 1}}A.A'.\left( {{A^{ - 1}}} \right)'\,\,\,\,\,\,$$ $$\left\{ {} \right.$$ as $$\,\,\,\,\,\,$$ $$AA' = A'A$$ $$\left. \, \right\}$$
$$ = I\left( {{A^{ - 1}}A} \right)'$$
$$ = I.I = {I^2} = I$$
Comments (0)
