Originally Posted by

**Runty** This problem I have partially solved, but I'm having issues concerning format and proper proofs. The question is listed below.

Let $\displaystyle A$ and $\displaystyle B$ be orthogonal $\displaystyle n\times n$ matrices. Determine which of the following are orthogonal (provide reasons for answers).

i) $\displaystyle A^{-1}$ (my answer is below, but I'm not sure if it is concrete enough)

Let $\displaystyle A=[a_{jk}]$. Since $\displaystyle A$ is orthogonal, then $\displaystyle A^{-1}=A^T=[a_{kj}]$. Therefore, $\displaystyle A^{-1}$ is orthogonal.

ii) $\displaystyle A-B$

(I'm afraid I'm not sure about this one)

If $\displaystyle AC$ is also an orthogonal $\displaystyle n\times n$ matrix, must $\displaystyle C$ be orthogonal?

I've inferred that $\displaystyle C$ is orthogonal, using the following three equivalent statements for some $\displaystyle n\times n$ matrix $\displaystyle Q$:

a. $\displaystyle Q$ is orthogonal.

b. $\displaystyle ||Qx||=||x||$ for every $\displaystyle x\in R^n$

c. $\displaystyle Qx\cdot Qy=x\cdot y$ for every $\displaystyle x,y\in R^n$

The issue I'm having is SHOWING how C is orthogonal without 'begging the question' (without using circular logic).

Can anyone help me with this?