We have
[tex]\det(AB)=\det A\det B[/tex]
for any two matrices [tex]A,B[/tex], and
[tex]\det A^{-1}=\dfrac1{\det A}[/tex]
[tex]\det A^\top=\det A[/tex]
[tex]\det(kA)=k^n\det A[/tex]
where [tex]k[/tex] is a constant and [tex]n[/tex] is the size of the matrix [tex]A[/tex].
1. [tex]\det(AB)-\det(BA)=\det A\det B-\det B\det A=0[/tex]
2. [tex]\det(A^2)+\det(B^2)=(\det A)^2+(\det B)^2=34[/tex]
3. [tex]\det(2A^3B)=2^4\det(A^3B)=16(\det A)^3\det B=2160[/tex]
4. [tex]\det(A^\top BA)=\det A^\top\det B\det A=45[/tex]
5. [tex]\det(B^{-1}AB)=\dfrac1{\det B}\det A\det B=\det A=3[/tex]
