How Do You Solve Determinants for Matrices A and B with Given Conditions?

[Whiz]

Homework Statement

If A and B are 3 x 3 matrices satisfying det(2A(B^-1)) = -20 and det((A^2)(B(transpose))) = 50, find det(A) and det(B)

Homework Equations

None

The Attempt at a Solution

I'm not quite sure how to do the question at all. I've just been guessing the determinants and seeing if it satisfies the two equations. I found that det(A) = -5 and det(B) = 2, but I just guessed.

How can I work this out?

 

[HallsofIvy]

Use the facts that det(AB)= det(A)det(B), and that det(A transpose)= det(A) so that 2det(A)/det(B)= -20 and (det(A)²)det(B)= 50 so that you have two equations to solve for det(A) and det(B). (And det(A)= -5, det(B)= 2 is not correct. 2(-5)/(2)= -5, not -20.)

Can you solve 2x/y= -20 and x²y= 50?

And, the word is "determinant", not "determinate".

 

[Whiz]

Thanks for the quick reply.

I'm not sure that's how you solve this question because the answer in the book is det(A) = 5 and det(B) = 2, which would make my answer wrong as well.