设A为n阶矩阵,满足A2=A,设A为n阶矩阵,满足A2=A,试证:r(A)+r(A+I)=n在下感激不尽!

Question

设A为n阶矩阵,满足A2=A,设A为n阶矩阵,满足A2=A,试证:r(A)+r(A+I)=n
在下感激不尽!

Answer 1

（结论应该是rank(A)+rank(A-I)=n,否则是错的.例：取A=I,则A^2=I=A,但rank(A)+rank(A+I)=rank(I)+rank(2I)=n+n=2n）证法一：令U={x∈R^n|Ax=0}为A的解集,则dim(U)=n-rank(A)；令V={x∈R^n|Ax=x}={x∈R^n|(A-I)x=0}为...