Calculating the determinant of $-2A^{-1}$ given the determinant of $A$.

Question

If $A$ is a square matrix or size $3$, where $\left | \ A \ \right| = -3$

How do you calculate something like $$ \left | -2A^{-1} \ \right |$$

?

Well, for starters, I believe that the determinant of a matrix is the same as the determinant of its inverse, so I guess the $A^{-1}$ is not a big problem. But what happens when you multiply a matrix by some scalar ($-2$)? Do I literally just multiply the determinant (which is $-3$) by $-2$?

Answer 1

Some rules about the determinant: if $A$ and $B$ are two square matrices with the same dimension $d$, $$\det(AB)=\det(A)\det(B).$$ In particular, since $\det(I)=1$, the determinant of the inverse of a matrix $A$ is the inverse of the determinant of $A$.

For the multiplication by a scalar, $$\det(\lambda A)=\lambda^d\det(A)$$ since the determinant is a multilinear form.

Answer 2

First, $\det(A^{-1})$ is not equal to $\det(A)$, but rather $\frac{1}{\det(A)}$.

Second, if $A$ is an $n\times n$ matrix and $k$ a scalar, then $\det(kA)=k^n\det(A)$.