Solving A^7 Using Matrix Multiplication: A 3x3 Example

In summary, the author is asking for help with a MATLAB function to multiply two matrices, but is having difficulty understanding the for loop.
  • #1
robbondo
90
0

Homework Statement




I can't figure out how to latexa 3X3 matrix so here's my ghetto method

A =
0 0 -1
0 2 0
0 0 1




What is

Homework Equations



I'm trying to find [tex] A^{7} [/tex]

The Attempt at a Solution



I'm assuming there's some type of shortcut to get this... So I did a squared and then multipled a squared times a squared to get a to the fourth... I was about to multiply A times a squared and then that times a to the fourth but I didn't think that was ok to do since multiplication isn't commutative and I don't know which order to multiply... Is my method ok? Any other suggestions?
 
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  • #2
you should look for which numbers stay the same regardless of the number of times you multiply it. There are some other elements that stay the same. Which?
 
Last edited:
  • #3
robbondo said:
I didn't think that was ok to do since multiplication isn't commutative
In general it's not... but can you prove that it is in this case?


I don't know which order to multiply...
What does the definition of exponentiation say?
 
  • #4
Oops I screwed up on the original posting...

so the matrix is

0 0 -1
0 2 0
2 0 0

And I don't really see any repetition at all which was what I was hoping would happen... For A squared I got

-2 0 0
0 4 0
0 0 -2

Then for A^4 I get

4 0 0
0 16 0
0 0 4

So there's no really observable pattern that I can see except for the fact that A^2 and A^4 so A^4 is all the values from A^2 squared... But now when I go to start multiplying things that aren't the same I don't know which order to do it in...
 
  • #5
I don't know what the deff. of exponentiation is... Exponentiation for matrix's or in general? This is the very beginning of my lin alg class...
 
  • #6
robbondo said:
I don't know what the deff. of exponentiation is... Exponentiation for matrix's or in general? This is the very beginning of my lin alg class...
Exponentiation for matrices. Your text should either precisely explain the order of operations, or indicate why the order shouldn't matter. (And if it doesn't, it ought to. :tongue:)

The order, in fact, doesn't matter -- and I think it would be a good exercise for you to try and figure out why. Start with an easy case -- e.g. show that

M(MM) = (MM)M

for any square matrix M.
 
  • #7
In general. Well, for nonnegative integer exponents anyway.
 
  • #8
I guess that makes sense since it's the same matrix. My sucky book doesn't have anything under exponentiation in the index but it may be in there, just not in the beginning sections were in right now. Thanks for the help, I finished the problem.
 
  • #9
Your "sucky" book probably assumed that you had taken basic algebra, at least enough to know what "exponentiation" meant, before taking linear algebra.
 
  • #10
HallsofIvy said:
Your "sucky" book probably assumed that you had taken basic algebra, at least enough to know what "exponentiation" meant, before taking linear algebra.
Linear algebra is the first time most people encounter a non-commutative multiplication operator, and are expected to be paying careful attention to their products to make sure they don't accidentally misapply a theorem of commutative arithmetic. I think it's fair to demand that an introductory textbook at least state explicitly that the intuitive definition of exponentiation as "repeated multiplication" is well-defined, and can be computed in any order.
 
  • #11
hello,
i don't understand about matrix multiplication especailly in writing an M-files codes.Hope anyone can help me..:(
 
  • #12
I think most of the people on this forum are very helpful if you are able to ask a specific question... Are you asking how to make an m file that will multiply two matrices? If so, it is a reatively long process. Is there a specific part you are having difficulties with?
 
  • #13
actually,i was asked to write a MATLAB function that multiplies matrices A and B using nested for loops...since i don't really understand about for loops and while,i can't write it.
 

What is matrix multiplication?

Matrix multiplication is a mathematical operation that takes two matrices as inputs and produces a new matrix as output. It involves multiplying corresponding elements in each row of the first matrix with the corresponding elements in each column of the second matrix, and then summing the results.

What are the requirements for two matrices to be multiplied together?

The number of columns in the first matrix must be equal to the number of rows in the second matrix. This is known as the compatibility condition for matrix multiplication.

What is the difference between matrix multiplication and regular multiplication?

In regular multiplication, we multiply two numbers together to get a single number as the result. In matrix multiplication, we multiply two matrices together to get a new matrix as the result. Additionally, matrix multiplication is not commutative, meaning changing the order of the matrices being multiplied will result in a different output.

What is the purpose of matrix multiplication in scientific applications?

Matrix multiplication is used in many scientific fields, including physics, engineering, and computer science. It is often used to solve systems of linear equations, transform coordinates, and perform calculations in linear algebra. It is also an important tool for data analysis and machine learning.

Can any two matrices be multiplied together?

No, not all matrices can be multiplied together. As mentioned earlier, the number of columns in the first matrix must be equal to the number of rows in the second matrix for them to be compatible for multiplication. Additionally, the dimensions of the resulting matrix will depend on the dimensions of the two input matrices and may not always be possible or meaningful in certain contexts.

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