Let Ax=b be a linear system with 5 equations and 5 unknowns in matrix form. The inverse of A does not exist. Which of the following statements is definitely always true?

Question

BlackTom AI · Accepted Answer

We start by restating the problem in our own words to set the stage: A is a 5x5 matrix, the system Ax = b has 5 equations and 5 unknowns, and A is singular (its inverse does not exist). We need to determine which statement is definitely always true given these conditions.

Option a: "The system has exactly two solutions." This is not guaranteed. When A is singular, the system cannot have a unique solution, but it can have zero, infinitely many, or (in some degenerate setups) technically many, but not exactly two. The claim of exactly two solutions would require a single nontrivial constraint that yields a precise finite count, which isn’t supported by linear systems theory for singular matrices.

Option b: "The system has infinitely many solutions." This can be true for some right-hand sides b (specifically, when b lies in the column space of A, the homogeneous part yields nontrivial solutions and you get infinitely many solutions). However, it is not guaranteed for all b; there exist b for which the system is inconsistent (no solutions). Therefore this is not definite.

Option c: "The system has no solutions." This can also be true for some b (when b is not in the column space of A), but again, it is not guaranteed for all b. Since A is singular, some right-hand sides produce no solution, but others produce infinitely many; thus this is not definite.

Option d: "The system does not have a unique solution." This is the statement that must hold for every possible b when A is singular. A singular matrix does not have full rank, so the linear system cannot have a unique solution for any b. While the exact number of solutions can vary (zero or infinitely many), the absence of a unique solution is guaranteed in all cases. Therefore this option is definitively true under the given condition.

In summary, among the provided choices, only option d is universally true whenever A is non-invertible for a 5x5 system, since a singular matrix cannot yield a unique solution for Ax = b regardless of b."

Let Ax=b be a linear system with 5 equations and 5 unknowns in matrix form. The inverse of A does not exist. Which of the following statements is definitely always true?单项选择题

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