Explanation
Solution
Step 1: Find the determinant of the coefficient matrix (Δ)
For a system of linear equations to be inconsistent or have infinitely many solutions, the determinant of the coefficient matrix must be zero.
Δ=λ11amp;1amp;λamp;1amp;1amp;1amp;λ=0
Expanding the determinant:
λ(λ−1)(λ+1)−(λ−1)−(λ−1)=0
Taking (λ−1) common:
So, λ=1 or λ=−2.
Step 2: Check for Inconsistency
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Case 1: If λ=1
The equations become:
x+y+z=1
x+y+z=1
x+y+z=1
These are identical planes, meaning there are infinitely many solutions. Therefore, λ=1 is not in set S.
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Case 2: If λ=−2
The equations become:
−2x+y+z=1 --- (i)
x−2y+z=1 --- (ii)
x+y−2z=1 --- (iii)
Adding all three equations: (−2+1+1)x+(1−2+1)y+(1+1−2)z=1+1+1
0x+0y+0z=3⇒0=3, which is impossible.
Thus, the system is inconsistent. So, S={−2}.
Step 3: Calculate the required value
We need to find ∑λ∈S(∣λ∣2+∣λ∣). Since S only contains −2:
Correct Option: (3) 6
