Finding (A²B²)⁻¹ for matrices A and B

[fredrick08]

Homework Statement

hi, i have 2 nxn matrices A and B i found that (AB)^-1=A^-1xB^-1 but how do i do
(A^2B^2)^-1=?

please i really need some help here I am kinda lost i have
AA^-1=I
AI=A
(AB)(A^-1xB^-1)=I

anyone have any ideas?

 

[HallsofIvy]

Homework Statement

hi, i have 2 nxn matrices A and B i found that (AB)^-1=A^-1xB^-1 but how do i do
(A^2B^2)^-1=?

Well, it's really unfortunate that you "found" that because it is not true!

please i really need some help here I am kinda lost i have
AA^-1=I
AI=A
(AB)(A^-1xB^-1)=I

No! you can't cancel BA^-1! That's why (AB)^-1 is NOT A^-1B^-1.

In order to be able to "cancel" A with A^-1 or B with B^-1, because multiplication of matrices is not commutative, you have to reverse the order of multiplication. (AB)(B^-1A^-1)= A(BB^-1)A^-1= A(I)A^-1= AA^-1= I. (AB)^-1= B^-1A^-1.

One thing you could do is think of (A²B²)^-1 as (UV)^-1= V^-1U^-1 where U= A² and V= B²- that is (A²B²)^-1= (B²)^-1(A^-1)²= (A^-1)²(B^-1)².

Similarly, (AABB)(B^-1B^-1A^-1A^-1)= AAB(BB^-1)B^-1A^-1A^-1= AA(B^-1B^-1)A^-1A^-1= A(AA^-1)A^-1= AA^-1= I

anyone have any ideas?

 

[fredrick08]

oh yes... sorry that was a typo it should of been B^-1A^-1 because of socks and shoes formula... sorry, ok yes i understand what u mean, yer i kept thinking of it as A^2 and B^2 instead of AA and BB just got myself confused... thankyou heaps!