Determining Solutions | The Significance of Zeros in the Reduced Row Echelon Form of a Matrix

if there is no solution, the reduced row echelon form of C has a row of zeros

In linear algebra, the reduced row echelon form (RREF) of a matrix is a special form obtained by performing row reduction operations on the given matrix

In linear algebra, the reduced row echelon form (RREF) of a matrix is a special form obtained by performing row reduction operations on the given matrix. The RREF has a few key properties that can help determine the existence of solutions to a system of linear equations.

If a system of linear equations is inconsistent, meaning it has no solution, it implies that there is a contradiction in the given equations. In terms of matrices, this would correspond to having a row of zeros in the RREF since it represents an equation of the form 0 = a, where ‘a’ is a non-zero constant.

To see why this is the case, let’s consider a system of linear equations represented by the augmented matrix [A|B]. Performing row reduction operations on this matrix until we obtain the RREF [C|D], we can observe the following possibilities:

1. If there is a row in the RREF of the form [0 0 … 0|c], where ‘c’ is non-zero, this implies 0 = c, which is a contradiction. Therefore, the system is inconsistent and has no solution.

2. If there is no row of zeros in the RREF, then every variable in the system will have a corresponding pivot column in the RREF. In this case, the system is consistent and has at least one solution. The number of pivot columns will give you information about the dimension of the solution space (whether it is a unique solution or infinitely many solutions).

Therefore, if the reduced row echelon form of matrix C has a row of zeros, it indicates that there is no solution to the system of linear equations represented by matrix C.

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