The sum of distinct values of λ for which the system of equations
( λ - 1 )x + ( 3λ + 1 )y + 2λ z = 0
( λ - 1 )x + ( 4λ - 2 )y + ( λ + 3 )z = 0
2x + ( 3λ + 1 )y + 3( λ - 1 )z = 0
has non-zero solutions, is ________ .

Solution

Maths sir · 4 steps

Non-zero solutions mean the determinant is zero. Subtract row 1 from the others and it factorises.

1. Condition. A homogeneous system always has x = y = z = 0. It has a non-zero solution only when the coefficient determinant Δ is zero.
2. Row operations. Apply R₂ → R₂ - R₁ to get (0, λ - 3, 3 - λ) = (λ - 3)(0, 1, -1). Apply R₃ → R₃ - R₁ to get (3 - λ, 0, λ - 3) = (λ - 3)(-1, 0, 1). Take (λ - 3) out of each of these two rows.

   Δ = (λ - 3)² |λ - 1, 3λ + 1, 2λ; 0, 1, -1; -1, 0, 1| = (λ - 3)²[(λ - 1) + (3λ + 1) + 2λ] = 6λ(λ - 3)²

3. Solve. 6λ(λ - 3)² = 0 gives λ = 0 or λ = 3. The root λ = 3 is repeated, but it is still only one value.
4. So. The distinct values are 0 and 3, and their sum is 3.

The common slip: 6. Counting the repeated root λ = 3 twice gives 0 + 3 + 3 = 6, but the question asks for distinct values.

The idea to keep. For a homogeneous system, set the determinant to zero. Look for row differences that share a common factor, because pulling it out early saves a lot of expansion.

Answer from the NTA final answer key. Solution by PrepWiser.