Question

Which of the following statements is/are TRUE ?
Note: \(\R\) denotes the set of real numbers.

• A. There exist M ∈ \(\R^{3×3}\), p ∈ \(\R^3\), and q E R3 such that Mx = p has a unique solution and Mx = q has infinite solutions.
• B. There exist M ∈ \(\R^{3×3}\), p ∈ \(\R^3\), and q E R3 such that Mx = p has no solutions and Mx = q has infinite solutions.
• C. There exist M ∈ \(\R^{2×3}\), p ∈ \(\R^2\), and q ∈ \(\R^2\) such that Mx = p has a unique solution and Mx = q has infinite solutions.
• D. There exist M ∈ \(\R^{3×2}\), p ∈ \(\R^3\), and q ∈ \(\R^3\) such that Mx = p has a unique solution and Mx = q has no solutions.

Answer 1

The Correct Option is B, D

Option (B) states that there exist a matrix M ∈ \(\R^{3×3}\), a vector p ∈ \(\R^3\), and a vector q ∈ \(\R^3\) such that the equation Mx = p has no solutions, while the equation Mx = q has infinitely many solutions. Option (D) states that there exist a matrix M ∈ \(\R^{3×2}\), a vector p ∈ \(\R^3\), and a vector q ∈ \(\R^3\) such that the equation Mx = p has a unique solution, while the equation Mx = q has no solutions.