Which of the following statements are true? A. If AA is an m×nm×n matrix and if

Question

Which of the following statements are true? A. If AA is an m×nm×n matrix and if the equation Ax=bAx=b is inconsistent for some bb in RmRm, then AA cannot have a pivot position in every row. B. If the augmented matrix [[ AA bb ]] has a pivot position in every row, then the equation Ax=bAx=b is inconsistent. C. The equation Ax=bAx=b is consistent if the augmented matrix [[ AA bb ]] has a pivot position in every row. D. Every matrix equation Ax=bAx=b corresponds to a vector equation with the same solution set. E. A vector bb is a linear combination of the columns of a matrix AA if and only if the equation Ax=bAx=b has at least one solution. F. If the columns of an m×nm×n matrix AA span RmRm, then the equation Ax=bAx=b is consistent for each bb in RmRm.

Explanation / Answer

A.

True.

B.

False. The system is inconsistent if [A b] has a pivot in the last ("b") column. The system is consistent if the matrix A has a pivot in every row. If the augmented matrix has pivot position in each row, the equation Ax=b may or may not be consistent.

C.

False. If the co-efficient matrix has a pivot position in every row, then Ax=b is consistent. However, for an augmented matrix, if there is a pivot position in every row Ax=b may or may not be consistent.

D.

True. Ax is simply a notation, a corresponding vector equation can also be made.

E.

True.

F.

True. For each b, Ax=b has a solution, is a linear combination of columns of A span Rm and A has a pivot position in each row. If the columns span R^m, this says that every b in R^m is in the span of the columns, which is another way of saying that any b is a linear combination of the columns. Then the equation is consistent

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