# Math Help - Orthogonal problem

1. ## Orthogonal

If A and B are both orthogonal nxn matrices.

Can anyone show That AB is orthogonal...

Thanks

2. ## Orthogonal problem

If A and B are both orthogonal nxn matrices.

Can anyone show That AB is orthogonal...iv been stuck on this question for a while now

Thanks

3. Originally Posted by dopi
If A and B are both orthogonal nxn matrices.

Can anyone show That AB is orthogonal...iv been stuck on this question for a while now

Thanks
An $n\times n$ $A$ is orthogonal iff

$AA^T=I$

Now suppose $A$ and $B$ be orthogonal, and let
$C=AB$. Now $C^T=B^TA^T$, so:

$
CC^T=ABB^TA^T=AIA^T=AA^T=I
$

Hence $C$ is orthogonal.

RonL

4. Originally Posted by CaptainBlack
An $n\times n$ $A$ is orthogonal iff

$AA^T=I$

Now suppose $A$ and $B$ be orthogonal, and let
$C=AB$. Now $C^T=B^TA^T$, so:

$
CC^T=ABB^TA^T=AIA^T=AA^T=I
$

Hence $C$ is orthogonal.

RonL

Say if you add A and B (A+B)...would that be Orthognal?

5. Originally Posted by dopi
Say if you add A and B (A+B)...would that be Orthognal?
Not generally

RonL

6. From the definition, we have for A and B:

$
\begin{array}{l}
AA^{ - 1} = A^{ - 1} A = I \\
BB^{ - 1} = B^{ - 1} B = I \\
\end{array}
$

$
\left( {AB} \right)\left( {AB} \right)^{ - 1} = ABB^{ - 1} A^{ - 1} = AIA^{ - 1} = AA^{ - 1} = I
$

7. ## Don't make duplicate posts

Don't post the same question in two different fora.

I have merged these two because they both have responses.

RonL

8. Originally Posted by TD!
From the definition, we have for A and B:

$
\begin{array}{l}
AA^{ - 1} = A^{ - 1} A = I \\
BB^{ - 1} = B^{ - 1} B = I \\
\end{array}
$

$
\left( {AB} \right)\left( {AB} \right)^{ - 1} = ABB^{ - 1} A^{ - 1} = AIA^{ - 1} = AA^{ - 1} = I
$
This is true of any non-singlar square matrices $A$ and $B$!

RonL