Properties of Negative Exponents of Matrix

Theorem

Let $A$ : Square matrix

Then

• $A^{-1}$ is invertible and $(A^{-1}{)}^{-1}=A$
• $A^n$ is invertible and $(A^n{)}^{-1}=(A^{-n})=(A^{-1}{)}^n$
• For any scalar $k\ne 0$, $kA$ is invertible and $(kA{)}^{-1}=k^{-1}{A}^{-1}$