设n阶方阵A满足A^2=En 且 |A+En|不等于0，证明：A=En

A^2=AA=E===> A=A'=A^(-1)=A^* 并且A不为0或(-E)
因为E^2=E
===>A^2-E^2=0

===> (A+E)(A-E)=0

---> A=E

To your question:

If AB=0, and A has left invert, then B=0.
Because A^(-1)AB=A^(-1) 0=0 ====>B=0

用矩阵的秩

A^2 = En
===>(A+E)(A-E) = 0
===>0 = r[(A+E)(A-E)] >= r(A+E) + r(A-E)-n
====>r(A+E) + r(A-E) <= n
因为|A+En|不等于0，所以r(A+E) = n,所以 r(A-E) <= 0

所以r(A-E) = 0
故A - E = 0
A = E