GATE2025_CS1_Q23

Consider the given system of linear equations for variables x and y, where k is a real-valued constant. Which of the following option(s) is/are CORRECT?

x + ky = 1
kx + y = -1

Note: This is a Multiple Select Question (MSQ). One or more options may be correct.

✅ Final Answer:

The correct options are (A) and (D).

🔹 Short Answer:

The system has a unique solution when the determinant of the coefficient matrix is non-zero (i.e., 1-k² ≠ 0). This gives `k ≠ 1` and `k ≠ -1`. When `k=1`, the equations are inconsistent (x+y=1, x+y=-1), leading to **no solution**. When `k=-1`, the equations are dependent (x-y=1, -x+y=-1), leading to **infinite solutions**. Therefore, there is exactly one value of k (`k=1`) for no solution and exactly one value of k (`k=-1`) for infinite solutions.

🔸 Long Answer:

To analyze the number of solutions for this system of linear equations, we can use the determinant of the coefficient matrix or compare the ranks of the coefficient and augmented matrices.

Method 1: Using Determinants

The system can be written in matrix form Ax = b:

[ 1 k ] [ x ] = [ 1 ]
[ k 1 ] [ y ] = [ -1 ]

The determinant of the coefficient matrix A is:

det(A) = (1)(1) - (k)(k) = 1 - k²

• Exactly One Solution: A unique solution exists if and only if det(A) ≠ 0.
  1 - k² ≠ 0 => k² ≠ 1 => k ≠ 1 and k ≠ -1.
  Since this holds for an infinite number of real values, statement (C) is false.
• No Solution or Infinite Solutions: These cases occur if det(A) = 0.
  1 - k² = 0 => k² = 1 => k = 1 or k = -1.
  We must investigate these two specific values of k.

Case k = 1:
The system becomes: x + y = 1
x + y = -1 These equations are contradictory. The lines are parallel and distinct. There is **no solution**.

Case k = -1:
The system becomes: x - y = 1
-x + y = -1 Multiplying the second equation by -1 gives x - y = 1, which is identical to the first. The lines are coincident. There are **infinitely many solutions**.

Summary of Results:

• No solution: Exactly one value, k=1. This makes statement (A) **TRUE** and (B) **FALSE**.
• Infinite solutions: Exactly one value, k=-1. This makes statement (D) **TRUE**.
• Unique solution: For all real k except 1 and -1 (an infinite set). This makes statement (C) **FALSE**.

Bibliography:
- Strang, G. (2016). Introduction to Linear Algebra. Wellesley-Cambridge Press. (Chapter 1: Introduction to Vectors).
- Anton, H., & Rorres, C. (2013). Elementary Linear Algebra: Applications Version. Wiley. (Chapter 1: Systems of Linear Equations and Matrices).

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