Which of the following statements are true?

I. $(A^{-1})^{-1} = A$.

II. $(AB)^{-1} = A^{-1}B^{-1}$.

III. $(A')^{-1} = (A^{-1})'$.

IV. $(kA)^{-1} = kA^{-1}$ for a non-zero scalar $k$.

Explanation

1. 01

   I: $A$ and $A^{-1}$ are inverses of each other. III: transposing $AA^{-1} = I$ gives $(A^{-1})'A' = I$.

2. 02

   II is false: the order reverses, $(AB)^{-1} = B^{-1}A^{-1}$. IV is false: $(kA)^{-1} = \frac{1}{k}A^{-1}$.