4x4 matrix A where A^2 and A^3 do not equal A, but A^4 = A

[MDolphins]

The question is, find a 4x4 matrix A where A^2 and A^3 do not equal A, but A^4 = A.

So far, I have tried matrixes all involving 1's and 0's and -1's. I have also tried to use examples of nilpotent matrixes. However, I have not found anything close.

 

[Dick]

The question is, find a 4x4 matrix A where A^2 and A^3 do not equal A, but A^4 = A.

So far, I have tried matrixes all involving 1's and 0's and -1's. I have also tried to use examples of nilpotent matrixes. However, I have not found anything close.

Think about a matrix that represents a rotation.

 

[MDolphins]

I could come up with a rotation matrix that was 180 degrees but not one that was 360 degrees

 

[Mark44]

The question is, find a 4x4 matrix A where A^2 and A^3 do not equal A, but A^4 = A.

So far, I have tried matrixes all involving 1's and 0's and -1's. I have also tried to use examples of nilpotent matrixes. However, I have not found anything close.

In future posts, please use the homework template - it's there for a reason. Do not just delete its parts.

 

[Dick]

I could come up with a rotation matrix that was 180 degrees but not one that was 360 degrees

360 degrees is the identity matrix. You don't want that. What angle would work?

 

[MDolphins]

You are right. You'd want 270 degrees right?

 

[Dick]

You are right. You'd want 270 degrees right?

270 degrees is the same as -90. So if A rotates by -90, then A^2 rotates by -180, A^3 by -270 and A^4 by -360, which is the same as the identity again. So A^4 isn't A. It's I. Think about the angle a bit more.

 

[MDolphins]

Oh okay so would it be by 120 degree angles. A2 would be 120 a3 would be 240 A4 would be 360 or the initial A

 

[Dick]

Oh okay so would it be by 120 degree angles. A2 would be 120 a3 would be 240 A4 would be 360 or the initial A

Right answer, wrong numbers. A is 120, A^2 is 240, etc. And 360 isn't the initial A. The initial A is 120.

 

[MDolphins]

Okay so now I have the angles, how can I visualize where I want the numbers in the4 x 4 matrix

 

[Dick]

Okay so now I have the angles, how can I visualize where I want the numbers in the4 x 4 matrix

There are a lot of places to put them depending on how you want to do the rotation. Do you know how to write a 2x2 matrix that represents a rotation by 120 degrees in the x,y plane? If so, then just write a 4x4 matrix that rotates the first two coordinates by 120 degrees and leaves the other two coordinates fixed.

 

[Ray Vickson]

222222222

All I see of your response is "222222222". Is that truly your response, or is some kind of hardware/software glitch messing up your answer on my display?

 

[LCKurtz]

He fell into a 22222... rotational loop.

 

[jtbell]

Is that truly your response, or is some kind of hardware/software glitch messing up your answer on my display?

It's apparently his way of deleting all his posts in the thread. Fortunately you guys quoted enough of what he wrote so the thread still sort of makes sense!

 

[LCKurtz]

It's apparently his way of deleting all his posts in the thread. Fortunately you guys quoted enough of what he wrote so the thread still sort of makes sense!

So maybe it was on a take-home test or for extra credit or something like that and he is trying to cover his tracks? Bring out the horse whips...

 

[Curious3141]

His posts have certainly become full of "Number 2".