设 $A , B$ 为 $n$ 阶矩阵，记 $r( X )$ 为矩阵 $X$ 的秩，则下列不正确的是（ ）
A. $r\left(\begin{array}{lc} A & A B \\ O & A ^{ T } A \end{array}\right)=2 r( A )$
B. $r\left(\begin{array}{ll} A B & A \end{array}\right)=r\binom{ B A }{ A }$
C. $r\left(\begin{array}{ll} A & A A ^{ T } \\ O & A ^{ T } A \end{array}\right)=r\left(\begin{array}{cc} A & O \\ A ^{ T } A & A A ^{ T }\end{array}\right)$
D. $r\left(\begin{array}{ll} A & A B \\ O & B A \end{array}\right)=r\left(\begin{array}{cc} A & O \\ B A & A B \end{array}\right)$