How would I answer the following question about the determinant and adjugate of a matrix?

Question

"Let A be a 3 x 3 matrix with determinant 4. Then det(adj(${A^T)) = ?, det(adj(A^{-1}))}$ = ? and det(adj(4A)) = ?."

Are there any rules through which I can solve this? The fact that their are adjoints and determinants together is confusing me.

Any help will be highly appreciated!

Answer 1

The adjugate matrix satisfies:

1. $A^{-1}=\frac{1}{\det(A)} \text{adj}(A)$

Use (1) and the following properties of the determinant of a square matrix with dimensionality $n$:

• $\det(A^{\top})=det(A)$
• $\det(A^{-1})=\frac{1}{\det(A)}$.
• $\det(cA)=c^n \det(A)$