# Math Help - Linear Algebra Counterexample

1. ## Linear Algebra Counterexample

If AB + BA = 0 then A^2B^3 = B^3A^2 if it is true give a short proof if it is true give a counterexample

I'm sure that the implication stated above is wrong but im struggling to give a counter-example. Anybody care to give me a hand in forming a counter-example or would you like to point out that the statement is indeed true and give me a proof of it.

Thanks.

2. Hello
Originally Posted by ah-bee
If AB + BA = 0 then A^2B^3 = B^3A^2 if it is true give a short proof if it is true give a counterexample

I'm sure that the implication stated above is wrong but im struggling to give a counter-example. Anybody care to give me a hand in forming a counter-example or would you like to point out that the statement is indeed true and give me a proof of it.
$AB+BA=0 \Longleftrightarrow AB=-BA$

Multiplying both sides by $BAB$ yields $ABBAB=-BABAB$ which you can transform into $A^2B^3=B^3A^2$ using $AB=-BA$.

3. Originally Posted by ah-bee
If AB + BA = 0 then A^2B^3 = B^3A^2 if it is true give a short proof if it is true give a counterexample

I'm sure that the implication stated above is wrong but im struggling to give a counter-example. Anybody care to give me a hand in forming a counter-example or would you like to point out that the statement is indeed true and give me a proof of it.

Thanks.
Left multiply bothe sides by A and right multiply both sides by B^2:

$AB = -BA \Rightarrow A^2 B^3 = -A(BA)B^2$

Substitute -AB = BA into the right hand side:

$A^2 B^3 = -(AB)AB^2 \Rightarrow A^2 B^3 = BA A B^2$.

Keep playing that game:

$A^2 B^3 = BA (AB) B \Rightarrow A^2 B^3 = BA (-BA) B = -B(AB)(AB) = -B (-BA) (-BA)$

$= -B^2 (AB) A = -B^2 (-BA) A = B^3 A^2$.

4. Originally Posted by ah-bee
If AB + BA = 0 then A^2B^3 = B^3A^2 if it is true give a short proof if it is true give a counterexample

I'm sure that the implication stated above is wrong but im struggling to give a counter-example. Anybody care to give me a hand in forming a counter-example or would you like to point out that the statement is indeed true and give me a proof of it.

Thanks.
What makes you so sure that it is false?

Remember that matrix multiplication is associative (for matrices over the reals):

$A^2B^3 = AABBB = A(AB)BB$

$=A(-BA)BB = -ABABB = -(AB)(AB)B = -BABAB$

$= -B(AB)(AB) = -BBABA = -BB(AB)A$

$=BBBAA = B^3A^2$

5. darn... thats a lot of multiplication. never wouldve seen that tbh. thanks for all the help.