Assume the 4 times 8coefficient matrix of a system has four

Assume the 4 times 8-coefficient matrix of a system has four pivot columns. Is the system consistent? Prove or give a counterexample.

Solution

Each pivot column of the coefficient matrix has only one pivot in it, so the coefficient matrix has

four pivots. Each of these pivots have to be in a different row, so each of the four rows of coefficient

matrix have a pivot.That means that all four rows of A are nonzero, and that the leftmost nonzero entry of each row

of A is in one . That means A is in echelon form and has no pivots in the rightmost column, so the system

is consistent.