Let Ax = b be a system of linear equations where A is an m x n matrix, b is an…
1996
Let Ax = b be a system of linear equations where A is an m x n matrix, b is an m x 1 column vector, and x is an n x 1 column vector of unknowns. Which of the following is false?
- A.
The system has a solution if and only if A and the augmented matrix [A b] have the same rank
- B.
If m < n and b is the zero vector, then the system has infinitely many solutions
- C.
If m = n and b is a non-zero vector, then the system has a unique solution
- D.
The system will have only a trivial solution when m = n, b is the zero vector, and rank(A) = n
Attempted by 3 students.
Show answer & explanation
Correct answer: C
Option C is false. A square system Ax = b has a unique solution only when A is nonsingular, equivalently rank(A) = n. The condition m = n and b is non-zero does not by itself guarantee uniqueness; if A is singular, the system may have no solution or infinitely many solutions.
Option A is true by the rank consistency condition: Ax = b is consistent exactly when rank(A) equals rank([A b]). Option B is true because when m < n and b = 0, the homogeneous system has at least one free variable and hence infinitely many solutions. Option D is true because a square homogeneous system with rank(A) = n has only the trivial solution.