Final answer:
A 4 x 7 matrix with 4 pivot positions indicates full rank by rows and ensures there won't be a row of all zeros in the row echelon form, implying the system is consistent. The statement is therefore True.
Step-by-step explanation:
If a 4 x 7 matrix has 4 pivot positions, we can infer it refers to the matrix having full rank in terms of the number of its rows because it has as many pivots as it has rows. In the context of solving a system of linear equations, pivot positions typically correspond to leading 1's in the row echelon form of the matrix, indicating that for each row, there is a leading term during the elimination process.
For a matrix to be consistent, at least one solution must exist for the equation Ax = b, where A is the matrix, x is the vector of variables, and b is the outcome vector. In this scenario, because the matrix has a pivot in every row, this implies that no row will end up having all zeros without a corresponding zero in the outcome vector b (which would indicate an inconsistent system if it had a non-zero result). Therefore, regardless of b, the system is consistent because it implies that every variable corresponds to a pivot and thus can have at least one solution.
Hence, the statement that a 4 x 7 matrix with 4 pivot positions is consistent is True.