Let A$!=$I$\_($n$)$ be an n$\backslash$times n matrix such that A$^(2)=$A$,$ where I$\_($n$)$ is the identity matrix of order

n$.$ Which of the following statements is false?

Rank$($A$)+$Rank$($I$\_($n$)-$A$)=$n

The eigenvalues of A are each equal to $1.$

Trace$($A$)=$Rank$($A$).$

$($I$\_($n$)-$A$)^(2)=$I$\_($n$)-$A