How to find the order of a matrix?

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In summary, the Heisenberg group over a field is a 3x3 upper right triangle matrix where the entries on the main diagonals are all 1's. Its order is determined by raising this matrix to a power that results in the identity matrix. However, since the upper right triangle with all 1's in the main diagonal can never become the identity matrix, the Heisenberg group has infinite order.
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xsw001
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For example the Heisenberg group over Field.

H(F) is a 3x3 upper right triangle where the entries on the main diagonals are all 1's.

So by definition that I need to use this matrix and raise to the power where it becomes the identity matrix, then the number of that power would the order of H(F). But if I take the upper right triangle with all 1's in the main diagonal, it would never becomes identity matrix though no matter how many times I multiple them.

I couldn't find any examples from the material that I have. Am I doing something wrong here? Any suggestions?
 
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If you have [tex]g^m\ne1[/tex] for every positive integer m, you say that g has infinite order.
 

1. What is the order of a matrix?

The order of a matrix refers to the number of rows and columns in the matrix. It is denoted by m x n, where m is the number of rows and n is the number of columns.

2. How do I find the order of a matrix?

To find the order of a matrix, simply count the number of rows and columns. For example, if a matrix has 3 rows and 4 columns, its order would be 3 x 4.

3. What is the significance of the order of a matrix?

The order of a matrix is important because it determines the dimensions of the matrix and affects how it can be manipulated and used in mathematical operations.

4. Can the order of a matrix be changed?

No, the order of a matrix is fixed and cannot be changed. Adding or removing rows or columns would result in a different matrix.

5. How does the order of a matrix affect matrix multiplication?

In matrix multiplication, the number of columns in the first matrix must match the number of rows in the second matrix. Therefore, the order of the matrices must be compatible for multiplication to be possible.

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