Understanding the Significance of a Row of Zeros in a Matrix | Implications for Solution Quantity and Consistency

if A has a row of zeros, there is more than one solution

When a matrix A has a row of zeros, it means that one of the rows in the matrix consists entirely of zeros

When a matrix A has a row of zeros, it means that one of the rows in the matrix consists entirely of zeros. In this case, there are two possibilities:

1. If the corresponding entry in the augmented column (if present) is also zero, then there are infinitely many solutions to the system of equations represented by the matrix A. This is because the row of zeros represents a redundant equation that does not provide any new information.

2. If the corresponding entry in the augmented column is non-zero, then there will be inconsistencies in the system of equations, and there will be no solution. This is because the row of zeros contradicts the non-zero entry in the augmented column.

It’s important to note that the number of solutions is not determined solely by the presence of a row of zeros, but also by other factors such as the values in the augmented column or the relationships between the other rows and columns in the matrix.

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