True or false: Each matrix is row equivalent to one and only one reduced echelon matrix.
1
True.
Row operations are reversible, and each row operation can be represented by a matrix operation. Therefore, any sequence of row operations can be represented by a sequence of matrix operations.
Starting from a given matrix, performing a sequence of row operations is equivalent to multiplying the matrix by a sequence of elementary matrices. Thus, row equivalent matrices can be represented as products of the original matrix and a sequence of elementary matrices.
Reduced row echelon form is unique for each matrix, so any matrix that is row equivalent to the original matrix can be transformed into the same reduced row echelon form matrix by applying the same sequence of elementary matrices. Therefore, each matrix is row equivalent to one and only one reduced echelon matrix.
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