1. ## Orthogonal Matrix problems

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

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

i) $A^{-1}$ (my answer is below, but I'm not sure if it is concrete enough)
Let $A=[a_{jk}]$. Since $A$ is orthogonal, then $A^{-1}=A^T=[a_{kj}]$. Therefore, $A^{-1}$ is orthogonal.

ii) $A-B$

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

I've inferred that $C$ is orthogonal, using the following three equivalent statements for some $n\times n$ matrix $Q$:
a. $Q$ is orthogonal.
b. $||Qx||=||x||$ for every $x\in R^n$
c. $Qx\cdot Qy=x\cdot y$ for every $x,y\in R^n$

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?

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

i) $A^{-1}$ (my answer is below, but I'm not sure if it is concrete enough)
Let $A=[a_{jk}]$. Since $A$ is orthogonal, then $A^{-1}=A^T=[a_{kj}]$. Therefore, $A^{-1}$ is orthogonal.

ii) $A-B$

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

I've inferred that $C$ is orthogonal, using the following three equivalent statements for some $n\times n$ matrix $Q$:
a. $Q$ is orthogonal.
b. $||Qx||=||x||$ for every $x\in R^n$
c. $Qx\cdot Qy=x\cdot y$ for every $x,y\in R^n$

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?
i) $I=AA^T=(A^T)^T A^T.$ so $A^T=A^{-1}$ is orthogonal.

ii) not necessarly. for example we might have $A=B$.

iii) yes because we have $CC^T=A^TACC^TA^TA=A^TAC(AC)^TA=A^TIA=A^TA=I.$

3. Originally Posted by NonCommAlg
i) $I=AA^T=(A^T)^T A^T.$ so $A^T=A^{-1}$ is orthogonal.

ii) not necessarly. for example we might have $A=B$.

iii) yes because we have $CC^T=A^TACC^TA^TA=A^TAC(AC)^TA=A^TIA=A^TA=I.$
Thanks for those answers. Are you absolutely sure about the first one? It seems a little... incomplete to me. Not wrong, but a tad lacking in substance.