Question

线代中证明A,B是n阶方阵，（A-B)(A+B)=A^2-B^2的充要条件是AB=BA

还有一个哈，就是证明(A+B)^2=A^2+2AB+B^2的充要条件是AB=BA
谢谢哈~

Answer 1

充分条件
a^2=a
a^2=0.5(b+e)*0.5(b+e)=0.25(b+e)(b+e)=0.25(b^2+2b+e)=0.5(b+e)
b^2+2b+e=2(b+e)
得
b^2=e
必要条件
a=0.5(b+e)得2a-e=b
两侧都乘以自身
(2a-e)(2a-e)=b^2
4a^2-4a+e=b^2=e
得4a^2-4a=o(0矩阵)得a^2=a

Answer 2

(A-B)(A+B)=A^2-B^2的充要条件是AB=BA

证明：

1) "==>"
(A-B)(A+B) = AA+AB-BA-BB = AA-BB --> AB-BA=0

2) "<=="
AB=BA --> AB-BA=0 -->
AA-BB = AA+AB-BA-BB = A(A+B)-B(A+B) = (A-B)(A+B)

Answer 3

(A-B)(A+B)=A^2-B^2的充要条件是AB=BA
证明：
1)
"==>"
(A-B)(A+B)
=
AA+AB-BA-BB
=
AA-BB
-->
AB-BA=0
2)
"<=="
AB=BA
-->
AB-BA=0
-->
AA-BB
=
AA+AB-BA-BB
=
A(A+B)-B(A+B)
=
(A-B)(A+B)