If A is a [3 3] non-singular matrix such that AA'=A'A and B = A^ - 1A', then BB' equals:

Question

If $A$ is a $$3 \times 3$$ non-singular matrix such that $AA'=A'A$ and

$B = {A^{ - 1}}A',$ then $BB'$ equals:

Accepted Answer

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

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

$$ = {A^{ - 1}}A.A'.\left( {{A^{ - 1}}} \right)'\,\,\,\,\,\,$$ $$\left\{ {} \right.$$ as $$\,\,\,\,\,\,$$ $AA' = A'A$ $$\left. \, \right\}$$

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

$= I.I = {I^2} = I$

Solutions · 1