Which of the following statements is/are TRUE? Note: ℝ denotes the set of real…
2024
Which of the following statements is/are TRUE?
Note: ℝ denotes the set of real numbers.
- A.
There exist 𝑴 ∈ ℝ3×3 , 𝒑 ∈ ℝ3 , and 𝒒 ∈ ℝ3 such that 𝑴𝐱 = 𝒑 has a unique solution and M𝐱 = 𝒒 has infinite solutions.
- B.
There exist 𝑴 ∈ ℝ3×3 , 𝒑 ∈ ℝ3 , and 𝒒 ∈ ℝ3 such that 𝑴𝐱 = 𝒑 has no solutions and M𝐱 = 𝒒 has infinite solutions
- C.
There exist 𝑴 ∈ ℝ2×3 , 𝒑 ∈ ℝ2 , and 𝒒 ∈ ℝ2 such that 𝑴𝐱 = 𝒑 has a unique solution and M𝐱 = 𝒒 has infinite solutions.
- D.
There exist 𝑴 ∈ ℝ3×2 , 𝒑 ∈ ℝ3 , and 𝒒 ∈ ℝ3 such that 𝑴𝐱 = 𝒑 has a unique solution and M𝐱 = 𝒒 has no solutions.
Show answer & explanation
Correct answer: B, D
Concept
For a linear system 𝑴𝐱 = 𝒃 with 𝑴 ∈ ℝm×n, the rank–nullity theorem decides everything: 𝑴𝐱 = 𝒃 is consistent (has at least one solution) iff 𝒃 lies in the column space of 𝑴; and when it is consistent, the solution is UNIQUE iff the null space of 𝑴 is trivial — nullity = n − rank(𝑴) = 0 — otherwise every consistent 𝒃 gives INFINITELY many solutions. So whether unique / no-solution / infinite is even POSSIBLE is fixed by the shape and rank of 𝑴, while whether a particular 𝒃 actually achieves it is a separate question of consistency.
Application — check each shape
𝑴 ∈ ℝ3×3 (square): if 𝑴𝐱 = 𝒑 has a unique solution then nullity(𝑴) = 0, i.e. 𝑴 is invertible. An invertible 𝑴 sends EVERY vector to exactly one solution, so 𝑴𝐱 = 𝒒 for any 𝒒 is also uniquely solvable — never infinitely many. “Unique for 𝒑, infinite for 𝒒” is impossible for one square 𝑴.
𝑴 ∈ ℝ3×3 but SINGULAR (rank < 3, nullity ≥ 1): take 𝑴 = diag(1, 1, 0). Its column space is the plane {(a, b, 0)}. With 𝒑 = (0, 0, 1), outside the column space, 𝑴𝐱 = 𝒑 has no solution; with 𝒒 = (1, 1, 0), inside it, 𝑴𝐱 = 𝒒 has infinitely many solutions (x₃ is free). Both hold at once — achievable.
𝑴 ∈ ℝ2×3 (2 rows, 3 columns): rank(𝑴) ≤ min(2, 3) = 2, so nullity = 3 − rank(𝑴) ≥ 1 for EVERY choice of 𝑴 — the null space can never be trivial for this shape. A unique solution therefore never occurs for a 2×3 system, regardless of 𝒑 or 𝑴, so this behaviour is impossible before 𝒒 even enters the picture.
𝑴 ∈ ℝ3×2 (3 rows, 2 columns) with full column rank 2 (nullity = 0): take 𝑴 = [[1,0],[0,1],[0,0]]. With 𝒑 = (1, 1, 0), in the 2-dimensional column space, 𝑴𝐱 = 𝒑 has the unique solution 𝐱 = (1, 1); with 𝒒 = (0, 0, 1), outside that column space, 𝑴𝐱 = 𝒒 has no solution. Both hold at once — achievable.
Cross-check — summary
Statement | Shape of 𝑴 | Nullity fact | Achievable? |
|---|---|---|---|
A | 3×3 | needs nullity 0 (unique) and nullity > 0 (infinite) at once — contradictory | No |
B | 3×3, singular | nullity ≥ 1 — matches no-solution + infinite | Yes |
C | 2×3 | nullity ≥ 1 always — unique is impossible | No |
D | 3×2, full column rank | nullity = 0 — matches unique + no-solution | Yes |
Result: only the “no solution for 𝒑, infinite for 𝒒” behaviour (a singular 3×3 𝑴) and the “unique for 𝒑, no solution for 𝒒” behaviour (a full-column-rank 3×2 𝑴) are actually realizable. So the TRUE statements are the ones describing a singular 3×3 𝑴 and a full-column-rank 3×2 𝑴; the ones requiring a single square 𝑴 to be both invertible and singular, or requiring a 2×3 system to have a unique solution, are impossible.