# Thread: Matrices and multiplication commutativity

1. ## [Solved] Matrices and multiplication commutativity

In a proof I am studying a step uses a propriety of matrices I do not know, doing some tries it seems true but I can't find a way to prove it.

I'll use some almost-LaTeX

v \in R^{n \times 1}
u \in R^{1 \times n}
the propriety is:
(vu)^2 = (uv)(vu)

Where, of course, (uv) is a real number.

Probably I am just missing the obvious, but matrices multiplication isn't usually commutative. Those matrices are peculiar since made from products of vectors, but I can't see why it works.
Can you tell me how to prove it? (or prove it is wrong of course)

(btw, where can I find some instruction about how to use the [ math ][ /math ] mode?)

Thanks!

2. Originally Posted by ematb
In a proof I am studying a step uses a propriety of matrices I do not know, doing some tries it seems true but I can't find a way to prove it.

I'll use some almost-LaTeX

v \in R^{n \times 1}
u \in R^{1 \times n}
the propriety is:
(vu)^2 = (uv)(vu)

Where, of course, (uv) is a real number.

Probably I am just missing the obvious, but matrices multiplication isn't usually commutative. Those matrices are peculiar since made from products of vectors, but I can't see why it works.
Can you tell me how to prove it? (or prove it is wrong of course)
You only need Associativity.

By definition, $\displaystyle (vu)^2 = (vu)(vu)$

Now by associativity:

$\displaystyle (vu)^2 = (vu)(vu) = v(uv)u$

Now since (uv) is a real number, $\displaystyle v(uv) = (uv)v$, thus

$\displaystyle (vu)^2 = (vu)(vu) = v(uv)u = (uv)vu$

Hope it helps.

(btw, where can I find some instruction about how to use the [ math ][ /math ] mode?)Thanks!
Just put whatever you did between these tags.

Here is an example:

Type $$v \in R^{n \times 1}$$ to get the following symbol: $\displaystyle v \in R^{n \times 1}$

3. Thanks a lot, this explain everything.