Final answer:
Yes, the equation Ax = 0 has a nontrivial solution if A is a 3x2 matrix with two pivot positions. When there are more columns than rows, there will always be free variables, resulting in infinite solutions.
Step-by-step explanation:
Yes, the equation Ax = 0 has a nontrivial solution if A is a 3x2 matrix with two pivot positions. A pivot position is a leading 1 in a row of the matrix when it is in row-echelon form. In this case, since there are two pivot positions, it means there are two leading 1's in the row-echelon form of the matrix.
When there are more variables (columns) than equations (rows), as in this case, there will always be free variables. Free variables are variables that can take on any value. Since there are two free variables in this case, it means there are infinite solutions to the equation Ax = 0, with at least one nontrivial solution.
For example, if A is the matrix [1 0; 0 1; 0 0], the equation Ax = 0 has the nontrivial solution x = [0; 0], where both x1 and x2 can be any real number.