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Matrix : Statement Proving

Tell whether the below statements are true or false/ Proving needed

1. If A2=0 then A=0
2. If A2=E then A=E or A=-E
3. If AB=0 then BA=0
4. If A2B=BA2 then AB=BA
5. If AB=-BA then (AB)3=-A3B3

I've solved all but curious about 4."Whether" there is a clear proving that the statement is wrong.
Have some intent of discussing.
Reply 1
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persevere97
OP
P.S. This problem is from Korea Math1 : Matrix

(edited 10 years ago)
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Reply 2
Avatar for Smaug123
Smaug123
15
Original post by persevere97
Tell whether the below statements are true or false/ Proving needed

1. If A2=0 then A=0
2. If A2=E then A=E or A=-E
3. If AB=0 then BA=0
4. If A2B=BA2 then AB=BA
5. If AB=-BA then (AB)3=-A3B3

I've solved all but curious about 4."Whether" there is a clear proving that the statement is wrong.
Have some intent of discussing.

To prove that a statement is false, all you need to do is provide a counterexample. That is, find A,B such that A2B≠BA2A^2 B \not = B A^2.
Original post by persevere97
Tell whether the below statements are true or false/ Proving needed

1. If A2=0 then A=0
2. If A2=E then A=E or A=-E
3. If AB=0 then BA=0
4. If A2B=BA2 then AB=BA
5. If AB=-BA then (AB)3=-A3B3

I've solved all but curious about 4."Whether" there is a clear proving that the statement is wrong.
Have some intent of discussing.


Try A=(2001)B=(1221) A=\begin{pmatrix}2 & 0 \\0 & 1 \end{pmatrix} B= \begin{pmatrix}1 & 2 \\2 & 1 \end{pmatrix}
Reply 4
Avatar for persevere97
persevere97
OP
Original post by Smaug123
To prove that a statement is false, all you need to do is provide a counterexample. That is, find A,B such that A2B≠BA2A^2 B \not = B A^2.


First, thank you for your advice.
But I sometimes, am not able to find counter example.
Finding counter example = Only way?
Reply 5
Avatar for Smaug123
Smaug123
15
Original post by persevere97
First, thank you for your advice.
But I sometimes, am not able to find counter example.
Finding counter example = Only way?

It's by far the best way. It's much, much easier than proving a result to be false in general. I can't think of a single case where that's true - proofs of falsity of a statement *always* go "here is a counterexample", whether or not they explicitly give the counterexample. (For instance, if the false statement is "the function f(x)f(x) has no roots", we might say "by the intermediate value theorem, there is a value for which f(x)=0f(x) = 0", even if we don't say exactly what that value is.)

In example 4 above, A2B=BA2A^2 B = B A^2. I used the handy fact that {{1,1},{0,1}}^n = {{1,n},{0,1}} to get my answer, because if we use that as AA then we know A will send e1e_1 to e1e_1. Then we can pretty much pick any BB, because if we apply the equation to e1e_1, on the left hand side we have A2(B(e1))A^2( B(e_1)) and on the right hand side we have B(e1)B(e_1). That is, the only constraint we need on BB is that it doesn't send e1e_1 to e1e_1.

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