Question

In general, matrix multiplication is not commutative. Show by counterexample that this is true. Then give an example of when matrix multiplication is commutative, Explain

Answer 1

@satellite73

Answer 2

pick any matrix
make is simple, like a two by two matrix
multiply one way and then the other and see that you don't get the same thing

Answer 3

If I don't get the same thing it makes it not commutative or commutative?

Answer 4

\[ \left( \begin{array}{cc}
1 & 2 \\
3 & 4 \\
\end{array} \right)\times\left( \begin{array}{cc}
3 & 2 \\
1 & 4 \\
\end{array} \right) \] for example
try that one

Answer 5

then try \[\left( \begin{array}{cc}
3 & 2 \\
1 & 4 \\
\end{array} \right)\left( \begin{array}{cc}
1 & 2 \\
3 & 4 \\
\end{array} \right)\]

Answer 6

the result will not be the same
that shows that \(A\times B\neq B\times A\) which shows they are NOT commutative

Answer 7

= [5 10]
[13 22]

=[9 14]
[13 18]

They're different

Answer 8

OHH!

Answer 9

But if they're still the same answer if multiplied both ways means they're commuative?

Answer 10

yeah, that is what it means
commutative means for all \(A, B\) \(AB=BA\) not true for matrices

Answer 11

\[
\left(
\begin{array}{cc}
1 & 2 \\
2 & 2 \\
\end{array}
\right).\left(
\begin{array}{cc}
0 & 1 \\
1 & 1 \\
\end{array}
\right)=\left(
\begin{array}{cc}
2 & 3 \\
2 & 4 \\
\end{array}
\right)\\
\left(
\begin{array}{cc}
0 & 1 \\
1 & 1 \\
\end{array}
\right).\left(
\begin{array}{cc}
1 & 2 \\
2 & 2 \\
\end{array}
\right)=\left(
\begin{array}{cc}
2 & 2 \\
3 & 4 \\
\end{array}
\right)
\]

Answer 12

Okay I see ^.^ Thank you soo much! You good helpers only come out at night ;P

Now I just said need to find a multplication matrix that is commutative

Answer 13

hint
use the identity matrix
it commutes with everything

Answer 14

Take two diagonal matrices

Answer 15

or what @eliassaab said

Answer 16

Is it okay if I take to same matrix and put them as an example?

like this [1 2] [1 2]
[1 4] [1 4]

Answer 17

I can't find one @eliassaab

Answer 18

Found one:

[1 0] [3 0] [3 0]
[0 2] [0 4] = [0 8]

[3 0] [1 0] [3 0]
[0 4] [0 2] = [0 8]