3. If the statement is always true, circle True. If the statement is sometimes f

Question

3. If the statement is always true, circle True. If the statement is sometimes false, circle False. In each case, write a careful and clear justification or counterexample. (a) A homogeneous equation is always consistent. True False True False (b) The homogeneous equation Ax = 0 has the trivial solution if and only if the equation has at least one free variable. True False (c) If A is a 3 x 6 matrix with three pivot positions then Ax=b has at least one solution for every possible vector b in R3.

Explanation / Answer

3.

(a) True

A homogenous equation is always consistent. The trivial solution is always a solution.

(b) False

The trivial solution is always a solution to the equation Ax=0

(c) True

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