Is $\det(AB) =\det(BA)$

Question

Let $A, B$ be square matrices. I am having trouble proving if $$ \det(AB) = \det(BA) $$ is right or wrong. Can you please point me to the right direction?

Answer 1

For square matrices $A, B$:$$\det(AB) = \det A \cdot \det B = \det B\cdot\det A = \det(BA)$$

To better understand why $\det(AB) = \det A \cdot \det B$, see this post:

Answer 2

Yes if $A$ and $B$ are two square matrices with the same size then

$$\det(AB)=\det(A)\det(B)$$

Answer 3

Sketch of proof: You're probably overthinking this because matrices don't commute in general. It turns out it's irrelevant in this case. The key fact here is that even though matrices do not in general commute, the determinant is a real valued function. So det(A) and det(B) are real numbers and multiplication of real numbers is commutative regardless of how they're derived. So det(A)det(B) = det(B)det(A) regardless of whether or not AB=BA.So if A and B are square matrices, the result follows from the fact det (AB) = det (A) det(B).

You should be able to finish the proof,no problem now.