Suppose a 3 times 5 coefficient matrix for a system has thre

Suppose a 3 times 5 coefficient matrix for a system has three pivot columns. Is the system consistent? Why or why not? Suppose a system of linear equations has a 3 times 5 augmented matrix whose fifth column is a pivot column. Is the system consistent? Why or why not? Suppose the coefficient matrix of a system of linear equations has a pivot position in every row. Explain why the system is consistent.

Solution

1) yes because with 3 rows and 3 pivots the reduced row echelon form can\'t have a bottom row of
[0 0 0 0 1] this it can\'t be inconsistent.

2) No , a pivot in the 5th column means we have a row [0 0 0 0 1] this we have an inconsistent system

3) This means we must have a picot in the bottom row (i.e there\'s one in each row), and so we can\'t have a pivot in the last column of the augmented matrix. Thus it is consistent