Question

设A是n*m阶矩阵,B是m*n阶矩阵,如果En-AB是可逆矩阵,（E是单位矩阵）,证明：Em-BA也是可逆矩阵

Answer 1

证:因为

(E-BA)[E+B(E-AB)^-1A]

= E-BA+B(E-AB)^-1A-BAB(E-AB)^-1A

= E-BA+B(E-AB)(E-AB)^-1A

= E-BA+BA

= E.

所以 E-BA 可逆,且 (E-BA)^-1 = E+B(E-AB)^-1A.