Why Every Matrix Has a Unique Row Echelon Form | Defining and Proving the Uniqueness of Matrix Forms

Every matrix has a unique row echelon form.

To understand why every matrix has a unique row echelon form, we need to clarify the definitions of a matrix and row echelon form

To understand why every matrix has a unique row echelon form, we need to clarify the definitions of a matrix and row echelon form.

Definition 1: Matrix
A matrix is a rectangular array of numbers or symbols arranged in rows and columns. It is usually denoted by a capital letter. For example, we can have a matrix A with dimensions m x n, where m represents the number of rows and n represents the number of columns.

Definition 2: Row Echelon Form
A matrix is said to be in row echelon form if it satisfies the following conditions:
1. All rows consisting entirely of zeros are at the bottom.
2. The first nonzero element, called a leading entry, of each nonzero row is to the right of the leading entry of the previous row.
3. Any rows of zeros are at the bottom.
4. In a nonzero row, all entries below and above the leading entry are zeros.

Now, let’s prove that every matrix has a unique row echelon form:

Proof:
1. Start with an initial matrix A.
2. Choose any nonzero entry as the first pivot (leading entry) in the first column, let’s say at position (i, j).
3. Perform row operations, if necessary, to make all other entries below the pivot (in the same column) zero. These row operations can be achieved through elementary row operations such as scaling, swapping, or adding multiples of one row to another.
4. Move to the next column and repeat steps 2 and 3 with the remaining nonzero entries.
5. Continue this process until all columns have been processed.
6. At the end of this process, we obtain a matrix in row echelon form.

Now, let’s explain why this row echelon form is unique:
1. There is only one way to choose a leading entry in each column, which ensures that the leading entries are uniquely determined.
2. The row operations performed on the matrix are based on the values of the elements and their positions, so they are also uniquely determined.
3. As a result, the row echelon form obtained through this process is unique.

In conclusion, every matrix has a unique row echelon form because the process of determining the row echelon form is well-defined and produces the same result regardless of the path taken to get there.

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