Some properties of the determinant:
- For any two square matrices [tex]A,B[/tex], we have [tex]\det(AB)=\det A\det B[/tex].
- For an [tex]n\times n[/tex] matrix [tex]A[/tex], [tex]\det(kA)=k^n\det A[/tex] for [tex]k\in\Bbb R[/tex].
- [tex]\det(A^\top)=\det A[/tex]
- For an invertible matrix [tex]A[/tex], [tex]\det(A^{-1})=\dfrac1{\det A}[/tex].
So
a. [tex]\det(-A)=(-1)^5\det A=3[/tex]
b. [tex]\det(A^5)=(\det A)^5=243[/tex]
c. [tex]\det(-2A^\top)=(-2)^5\det(A^\top)=(-2)^5\det A=96[/tex]
d. Since [tex]\det A\neq0[/tex], the inverse [tex]A^{-1}[/tex] exists, so [tex]A^{-n}=(A^{-1})^n[/tex].
[tex]\det(A^{-3})=\det((A^{-1})^3)=(\det(A^{-1}))^3=\dfrac1{(\det A)^3}=-\dfrac1{27}[/tex]
