In which of the following type of matrix inverse does not exist always

For involutary matrix,  A2=I⇒A2=|I|⇒|A|2=1⇒|A|=±1  For idempotent matrix,A2=A⇒A2=|A|⇒|A|2=|A|⇒|A|=0 or 1   For orthogonal matrix, AAT=I⇒AAT=|I|⇒A∥AT=1⇒|A|2=1⇒|A|=±1   Thus if matrix A is idempotent it may not be invertible.