Unlocking The Power Of Matrices: The Unique Row Echelon Form

Every matrix has a unique row echelon form.

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This statement is true. The row echelon form of a matrix is obtained by performing elementary row operations on the matrix until it satisfies the following conditions:

1. All zero rows are at the bottom of the matrix.
2. The leading (leftmost) non-zero element of each row is 1.
3. The leading 1 of each row is to the right of the leading 1 of the row above it.
4. Any rows consisting entirely of zeros are grouped together.

The row echelon form is unique for every matrix, regardless of the method used to obtain it. This means that if two matrices have the same row echelon form, they are necessarily equal up to a permutation of rows and/or columns.

To see why this is true, suppose that we have two matrices, A and B, that have the same row echelon form R. We can use elementary row operations to transform A and B into R, and since A and B have the same row echelon form, the same set of operations will work for both. Therefore, after the transformation, A and B will also be equal up to a permutation of rows and/or columns.

Conversely, if two matrices A and B are equal up to a permutation of rows and/or columns, then their row echelon forms will be equal. This is because elementary row operations affect only the rows of a matrix, and therefore the row echelon form is invariant under column permutations.

Therefore, we can conclude that every matrix has a unique row echelon form.

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