Exercise 3.4.3

Answers

1.: We can check that $A^{'}$ is also reduced echelon form. So the number of nonzero rows in $A^{'}$ is the rank of $A$ . And the number of nonzero rows in $(A^{'} | b^{'})$ is the rank of $(A | b)$ . So if they have different rank there must contain some nonzero rows (actually only one row) in $(A^{'} | b^{'})$ but not in $A^{'}$ . This means the nonzero row must has nonzero entry in the last column. Conversely, if some row has its only nonzero entry in the last column, this row did not attribute the rank of $A^{'}$ . Since every nonzero row in $A^{'}$ has its corresponding row in $(A^{'} | b^{'})$ also a nonzero row, we know that two matrix have different rank.
2.: By the previous exercise we know that $(A^{'} | b^{'})$ contains a row with only nonzero entry in the last column is equivalent to that $A^{'}$ and $(A^{'} | b^{'})$ have different rank. With the help of Theorem 3.11 we get the desired conclusion.

jephianlin

2011-06-27 00:00