$f命题：$设$A in {M_{m imes n}}left( F ight),B in {M_{n imes m}}left( F ight),m ge n,lambda e 0$，则
[{
m{ }}left| {lambda {E_m} - AB}
ight| = {lambda ^{m - n}}left| {lambda {E_n} - BA}
ight|]
方法一：初等变换法
[left( {egin{array}{*{20}{c}}
{lambda {E_m}}&A\
B&{{E_n}}
end{array}}
ight) o left( {egin{array}{*{20}{c}}
{lambda {E_m} - AB}&0\
B&{{E_n}}
end{array}}
ight) o left( {egin{array}{*{20}{c}}
{lambda {E_m} - AB}&0\
0&{{E_n}}
end{array}}
ight)]
[left( {egin{array}{*{20}{c}}
{lambda {E_m}}&A\
B&{{E_n}}
end{array}}
ight) o left( {egin{array}{*{20}{c}}
{lambda {E_m}}&A\
0&{frac{1}{lambda }left( {lambda {E_n} - BA}
ight)}
end{array}}
ight) o left( {egin{array}{*{20}{c}}
{lambda {E_m}}&0\
0&{frac{1}{lambda }left( {lambda {E_n} - BA}
ight)}
end{array}}
ight)]