# should I use real numbers ??

• Sep 17th 2006, 05:59 AM
kgpretty
should I use real numbers ??
I have a maths question which reads:

Let M be the set of all real 2 * 2-matrices of the form:
(a 0)
(0 b)

Show that this set of matrices is closed under matrix multiplication, and that matrix multiplication is commutative on this set.

Should I use real numbers to solve this problem or use what the question gave??
• Sep 17th 2006, 06:01 AM
kgpretty
Re :should I use real numbers ??
I forgot to mention.. the attached image is the matrix given in the question.
• Sep 17th 2006, 06:12 AM
CaptainBlack
Quote:

Originally Posted by kgpretty
I have a maths question which reads:

Let M be the set of all real 2 * 2-matrices of the form:
(a 0)
(0 b)

Show that this set of matrices is closed under matrix multiplication, and that matrix multiplication is commutative on this set.

Should I use real numbers to solve this problem or use what the question gave??

Are you asking if you should replace a snd b with particular numbers?

If so the answer is no.

That M is closed under matrix multiplication is equivalent asking you
to prove that for any real numbers a, b, c, d, there exist two real numbers
e,f such that:

Code:

```[a 0] [c 0] = [e 0] [0,b] [0 d]  [0 f]```
and that multiplication is comutative on M equivalent asking you
to prove that for any real numbers a, b, c, d that:

Code:

```[a 0] [c 0] = [c 0] [a 0] [0,b] [0 d]  [0 d] [0 b]```
RonL
• Sep 17th 2006, 06:17 AM
kgpretty
Re: should I use real numbers ??
Quote:

Originally Posted by CaptainBlack
Are you asking if you should replace a snd b with particular numbers?

If so the answer is no.

That M is closed under matrix multiplication is equivalent asking you
to prove that for any real numbers a, b, c, d, there exist two real numbers
e,f such that:

Code:

```[a 0] [c 0] = [e 0] [0,b] [0 d]  [0 f]```
and that multiplication is comutative on M equivalent asking you
to prove that for any real numbers a, b, c, d that:

Code:

```[a 0] [c 0] = [c 0] [a 0] [0,b] [0 d]  [0 d] [0 b]```
RonL

• Sep 17th 2006, 06:40 AM
kgpretty
what exactly does a closed matrix mean? I'm confused.

I thought it meant.. in this case, that when I did the multiplication.. the resulting matrix would be the same size.
• Sep 17th 2006, 07:34 AM
ThePerfectHacker
Quote:

Originally Posted by kgpretty
what exactly does a closed matrix mean? I'm confused.

I thought it meant.. in this case, that when I did the multiplication.. the resulting matrix would be the same size.

Given a set S with an algebraic binary operation *
Take a subset R of S such that,
a*b in R for all a,b in R
Then we say R is closed under the binary operation *

--
Informally, closed mean whenever you take two numbers and you operate them you get the same number in the set.

For example,
If your matrix ended up being,
[0 a]
[b 0]
For a and b nonzero then your binary operation on this set would not have been closed because you have a diffrenet number (in this case a different matrix).

This is my 25:):)th Post
• Sep 17th 2006, 07:48 AM
kgpretty
I believe I've got it now..
Here's how I'm going to completely answer the question:

Code:

```Let a,b,c,d,e,f be real numbers: [a 0] [c 0] = [e 0] [0 b] [0 d] = [0 f] .: M is closed under matrix multiplication because the product of two M members is also in the M set. Let A = [a 0]         [0 a]     B = [b 0]         [0 b] AB = [ab  0]     [0  ab] BA = [ab  0]     [0  ab] AB = BA .: matrix multiplication is commutative on this set.```
• Sep 17th 2006, 07:51 AM
ThePerfectHacker
I think you meant
[a 0]
[0 b]
Not,
[a 0]
[0 b]

Anyways that is not the biggest problem, you need tos show that,
[ab 0]
[0 ba] is an element of these matricies.
Why?
Because a,b are real numbers and ab is closed under multiplication. That is the important step you omitted.
Because you need to show that,
[ab 0]
[0 ab]
is in the set which is true since ab is an element of the set.
• Sep 17th 2006, 07:58 AM
kgpretty
Quote:

Originally Posted by ThePerfectHacker
I think you meant
[a 0]
[0 b]
Not,
[a 0]
[0 b]

Anyways that is not the biggest problem, you need tos show that,
[ab 0]
[0 ba] is an element of these matricies.
Why?
Because a,b are real numbers and ab is closed under multiplication. That is the important step you omitted.
Because you need to show that,
[ab 0]
[0 ab]
is in the set which is true since ab is an element of the set.

``` Let A = [a 0]         [0 b]       B = [c 0]         [0 d]   AB = [ac  0]     [0  bd]   BA = [ca  0]     [0  db]```