simplify (matrices) (A B A^-1)^6

B

Bronn

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simplify (A B A^-1)^6 A and B being matrices where AB exists

the answer given is = A B^6 A^-1

I'm baffled how to get this answer, any help, please?



and one more question, I can't seem to find an answer anywhere online.

If you have (AB)^-1 it becomes B^-1 • A^-1

is the same true for (AB)^2, does it become B^2•A^2 ?

thanks
 
Bronn said:
simplify (A B A^-1)^6 A and B being matrices where AB exists

the answer given is = A B^6 A^-1

I'm baffled how to get this answer, any help, please?



and one more question, I can't seem to find an answer anywhere online.

If you have (AB)^-1 it becomes B^-1 • A^-1

is the same true for (AB)^2, does it become B^2•A^2 ?

thanks
Click to expand...

For the first question, start with a simpler case. What is (A B A^-1)^2? It is

(A B A^-1)(A B A^-1) = A B A^-1 A B A^-1 = A B (A^-1 A) B A^-1 = A B I B A^-1 = A B B A^-1 = A B^2 A^-1

Do you follow the steps? The same thing applies to any power.

For the second, the answer is no. The reason the inverse works as it does does not apply to just any multiplication.

Can you see why (AB)^-1 would be B^-1 A^-1? Recall that two matrices are inverses if their product is I. What is the product of AB and B^-1 A^-1 in this case? (It is closely related to the first question.)

I would think your textbook, or any site that introduces this idea, would surely explain why!
 
Bronn said:
simplify (A B A^-1)^6 A and B being matrices where AB exists

the answer given is = A B^6 A^-1

I'm baffled how to get this answer, any help, please?



and one more question, I can't seem to find an answer anywhere online.

If you have (AB)^-1 it becomes B^-1 • A^-1

is the same true for (AB)^2, does it become B^2•A^2 ?

thanks
Click to expand...
I'm going to (slightly) supplement Dr.Peterson's comment:
\(\displaystyle (AB)^2 = (AB)(AB) = ABAB\) which does not generally reduce to something simpler. Don't get caught up in rules like \(\displaystyle (AB)^{-1} = B^{-1} A^{-1}\). They are useful (especially this one) but don't them distract you from the definitions.

-Dan
 
Ohh ok I got it. Thanks, both!

It's course notes, the information is not explicitly stated but you can extrapolate it from the concepts. But easy to miss when you're fresh to it.
 
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