Answer
a)Ax = 0 has a nontrivial solution.
b)The equation Ax = b cannot have a solution for every possible b (in $R^4$).
Work Step by Step
a. When A is a 4×4 matrix with three pivot positions, the equation Ax = 0 has three basic variables and one free variable. So Ax = 0 has a nontrivial solution.
b. With only three pivot positions, A cannot have a pivot in every row, so by Theorem 4 in Section 1.4, the equation Ax = b cannot have a solution for every possible b $(in R^4)$.
