Orthogonal Matrix statements/proofs

Question

I am currently learning about Orthogonal matrices and have three statements that I can not figure out.

Here are three Proofs that I have written down but can not figure how to prove them:

If $Q$ is an orthogonal matrix, then $Q^{-1}$ is orthogonal.

If $Q$ is an orthogonal matrix, then $Q^T$ is orthogonal.

If $Q_1$ and $Q_2$ are Orthogonal matrices, then $Q_1Q_2$ is orthogonal.

I am having trouble understanding these. Can anyone clear these up?

Thanks!

For the first statement, I have:

$Q$ = $Q^T$

($Q$ - 1) $Q$ = $Q^T$ ($Q$ - 1)

($Q$ - 1)$Q$ = I

Answer 1

Suppose that $Q$ is orthogonal so that the rows and columns of the matrix are orthonormal. Then the $i^{th}$ row of $Q$ is the $i^{th}$ column of $Q^T$ so that $Q^T$ has orthonormal rows and columns as well. Hence it's orthogonal. The $ij^{th}$ entry of $QQ^T$ is the dot product of the $i^{th}$row of $Q$ with the $j^{th}$ column of $Q^T$. If $i=j$ this dot product is 1 by orthonormal if $i \neq j$ this dot product is 0 by orthonormal. Thus $QQ^T=I$. Since inverses are unique $Q^T=Q^{-1}$. Last $(Q_1Q_2)(Q_1Q_2)^T = Q_1Q_2Q_2^TQ_1^T =Q_1IQ_1^T=I$ showing that the product is and orthogonal matrix.

Answer 2

By one of the more or less usual definition:

$$Q\;\text{is orthogonal}\;\iff Q^t=Q^{-1}\implies \left(Q^t\right)^{-1}=\left(Q^{-1}\right)^t=\left(Q^{-1}\right)^{-1}=Q$$

From here you get at once the first two claims, and this will help you a lot (a lot!) also with the third one.