Let Ax=b be a linear system with 4 equations and 5 unknowns in matrix form. Which of the following statements is always true no matter what particular choice of A we are dealing with?

Question

BlackTom AI · Accepted Answer

Consider the setup: Ax = b where A is a 4×5 matrix (four equations, five unknowns).
Option a: The system has infinitely many solutions. This is not guaranteed. If the system is inconsistent (for example, b lies outside the column space of A), there are no solutions at all. If it is consistent, there are typically infinitely many solutions due to the underdetermined nature (more unknowns than independent equations), but the statement asks for a guarantee for all A and b, which is not satisfied because consistency is not ensured.

Option b: The system has exactly one solution. This cannot be guaranteed here. With five unknowns and at most four independent equations, you cannot pin down a unique solution in all cases. There will be either no solution or infinitely many solutions when the system is consistent.

Option c: The system has no solutions. Again, this is not guaranteed. There exist choices of A and b for which the system is consistent and has infinitely many solutions. Therefore, claiming no solutions always is false.

Option d: The matrix A is not invertible. Since A is 4×5 and not square, it does not have an inverse in the usual sense. Invertibility is defined for square matrices only, so this matrix cannot be invertible. This property holds for every possible A of size 4×5, regardless of the actual entries of A.

In summary, options a, b, and c hinge on the specific b and the rank of A, so they are not universally true. Option d, however, is universally true because a non-square matrix cannot be invertible in the standard linear-algebra sense.

Let Ax=b be a linear system with 4 equations and 5 unknowns in matrix form. Which of the following statements is always true no matter what particular choice of A we are dealing with?单项选择题

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