Suppose A and B are 4x4 matrices such that detA=-2 and detB=3 , find det [ A^3*(detB^-1)]

Question

Suppose A and B are 4x4 matrices such that detA=-2 and detB=3 , find det [ (A³)det(B^-1 ) ] 

 

I dont understand how to find det(detB^-1) . What is the property for this?

Answer 1

Use these properties of determinants of products of square matrices of the same order:

 

det(AB) = det(A)det(B) 

 

det(A^-1) = 1/det(A)

 

det(cA)=cⁿdet(A) for a scalar c and an n×n matrix

 

det[A³ * det(B^-1)] = ?

 

Note that det(B^-1) is just a scalar, which in this problem is 1/3

 

Then 

 

det[A³ * det(B^-1)] = det[(1/3)A³] = (1/3)⁴ det(A³) = (1/3)⁴ [det(A)]³ = (1/3)⁴ (-2)³

 

You can finish from here.