LetA and B be square matrices of the same dimensions. Suppose that
det(AB2) = 1
det(AB) = 3
Show that A and B are invertible and find their determinants.
I'm not even sure where to start with this!
LetA and B be square matrices of the same dimensions. Suppose that
det(AB2) = 1
det(AB) = 3
Show that A and B are invertible and find their determinants.
I'm not even sure where to start with this!
Start with these facts:
1) det(AB)= det(A)det(B)
2) A is invertible if and only det(A) is not 0.
3) xy= 0 if and only if either x= 0 or y= 0. So that if xy is not 0 then neither x nor y is 0.
det(AB)= 3 which is not 0 so neither det(A) nor det(B)= 0 and both A and B are invertible.
$\displaystyle det(AB^2)= det(AB)det(B)$ and you are told both det(AB) and det(AB^2).
So far what I have is det(AB)=det(A)det(B)=3 and det(AB(squared))= det(A)det(B(squared))
(square root of 1) = det(A)(square root (det(B)))
1 = det(A)(square root (det(B)))
1 = 1/3(square root of 9)
1 = 1/3(3)
1 = 1
det(AB)= det(A)det(B)
3 = 1/3(9)
3 = 3
But I'm not sure if this is right?