Which of the following statements are true?

I. A rectangular matrix that is not square has no inverse.

II. The inverse of a square matrix, when it exists, is unique.

III. Every square matrix has an inverse.

IV. If $B$ is the inverse of $A$, then $A$ is the inverse of $B$.

Explanation

1. 01

   I, II and IV are true: $AB = BA = I$ needs square matrices, two inverses $B$ and $C$ would give $B = B(AC) = (BA)C = C$, and the definition reads the same both ways.

2. 02

   III is false: $O$, or $\begin{bmatrix} 1 & 2 \\ 2 & 4 \end{bmatrix}$, has no inverse.