Question

SejaA2Mn(R);idempotemte, ou seja,A2=A:Mostre quedet A= 0 ou det A=1

Answer 1

Tese: $A^2=A$

Hipótese: $\det\,A=0\;\vee\;\det\,A=1$
 
$A^2=A\\\\(A\cdot\,A)=A\\\\A^{-1}\cdot(A\cdot\,A)=A^{-1}\cdot\,A\\\\(A^{-1}\cdot\,A)\cdot\,A=I\\\\I\cdot\,A=I\\\\\det(I\cdot\,A)=\det\,I\\\\\det\,I\cdot\det\,A=\det\,I\\\\1\cdot\detA=1\\\\\boxed{\det\,A=1}$
 
  Para provar que $\det\,A=0$ tome a matriz nula de ordem n!