Question

Suppose a coefficient matrix for a system has pivot columns. Is the system​ consistent? Why or why​ not? Choose the correct answer below. A. There is a pivot position in each row of the coefficient matrix. The augmented matrix will have columns and will not have a row of the form ​, so the system is consistent. B. There is a pivot position in each row of the coefficient matrix. The augmented matrix will have columns and will not have a row of the form ​, so the system is consistent. C. There is at least one row of the coefficient matrix that does not have a pivot position. This means the augmented​ matrix, which will have ​columns, could have a row of the form ​, so the system could be inconsistent. D. There is at least one row of the coefficient matrix that does not have a pivot position. This means the augmented​ matrix, which will have ​columns, must have a row of the form ​, so the system is inconsistent.

Answer 1

Answer:

A. There is a pivot position in each row of the coefficient matrix. The augmented matrix will have columns and will not have a row of the form ​, so the system is consistent.

Step-by-step explanation:

Based on Godel's second incompleteness theorem, a system is considered consistent only when the far-right column of the augmented matrix is not a pivot column.

However, given that every column of the coefficient matrix is a pivot column, then there are no top coefficients in the far-right column of the augmented matrix.

Hence, there is a pivot position in each row of the coefficient matrix. The augmented matrix will have columns and will not have a row of the form ​, so the system is consistent.