Matrix with Power of n: Evaluating Real Number Values | Homework Help

[MechaMZ]

Homework Statement

a and n could be any real number, please evaluate the value.

<br /> B =\left [<br /> \begin{array}{ccc}<br /> a&amp;1&amp;0\\<br /> 0&amp;a&amp;0\\<br /> 0&amp;0&amp;a<br /> <br /> \end{array}<br /> \right ]<br />ⁿ

I've no idea how to approach this question, because i have never came across the matrix with power not equals to -1.
could somebody give me some hints?

I've try to let n = -1 by using gauss jordan elimination, and let then elements become 1/a, 1/a² and etc. but this seems incorrect and not what the question asking for..

 

[Dick]

You can either expand it for small values of n and guess the pattern and prove it works by induction (if you have to). Or write the matrix as aI+J, where I is the identity and J is zero except for that off diagonal 1. Now, I and J commute, so you can apply the binomial theorem to (aI+J)^n.

 

[Mark44]

Are you sure that the problem says that n could be any real number? You're going to have a hard time raising your matrix to the power \pi, say.

 

[MechaMZ]

yeah, the question states the a and n is any real value. but why power of pi?

but I'm not very sure this question is asking me to find the n and a value or prove it could be done by any real value(for a and n)

 

[Mark44]

I understand that the problem is not asking you to find a and n, but if n can be any real number, then there is nothing that prevents n from being \pi. If so, what does it mean to raise a matrix to an irrational power? A more reasonable interpretation, IMO, would be that n is a positive integer, for which math induction would be one approach.

 

[HallsofIvy]

I'm with the others. If both a and n can be any real numbers, then for some combinations of a and n, the matrix won't even exist. For example, if a= 0 and n= -1, B^-1 will not exist. If a is negative and n = 1/2, you are going to run into complex numbers. If a can be any real number, and n any positive integer, the problem is relatively simple. Calculate B, B², B³, make a good guess at what the general form will be and, as Mark44 suggested, prove it by induction.

 

[Dick]

Boo hoo. Nobody likes my binomial theorem suggestion??

 

[Mark44]

I like it. I was just offering another possible slant.

 

[MechaMZ]

I'm with the others. If both a and n can be any real numbers, then for some combinations of a and n, the matrix won't even exist. For example, if a= 0 and n= -1, B^-1 will not exist. If a is negative and n = 1/2, you are going to run into complex numbers. If a can be any real number, and n any positive integer, the problem is relatively simple. Calculate B, B², B³, make a good guess at what the general form will be and, as Mark44 suggested, prove it by induction.

yeah, i like this solution =)
thanks all