Indicate whether the statement is always true or sometimes false. If it is somet

Question

Indicate whether the statement is always true or sometimes false. If it is sometimes false, please show a counterexample.

Indicate whether the statement is always true or sometimes false. If it is sometimes false, please show a counterexample. (a) A linear system of one equation in two unknown must be consistent. (b) If the number of equations in a linear system exceeds the number of unknowns, the system must be inconsistent (c) A homogeneous linear system of two equations in three unknowns has infinitely many solutions (d) If the reduced row echelon form of the augmented matrix for a linear system has a row of zero, then the system must have infinitely many solutions.

Explanation / Answer

Only the statement (b) is false. Here is a counterexample:

x+y=1

x-y=3

3x+y=5

The above system has 3 equations but 2 unknowns. However, it is consistent, as x=2 and y=-1 is a solution.

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