Match the following
x+y+z=2
x+y+λz=1
x+μy+2y=3
(a) μ=1,λ=1 (i) No solution
(b) μ=2,λ=3 (ii) Unique Solution
(c) μ=1,λ=0 (iii) Infinite Solutions
a-i; b – ii: c- iii
Let’s convert A into Echelon form of
A = ⎡⎢⎣11111λ1μ2⎤⎥⎦
R3→R3–R1,R2→R2–R1 ⎡⎢⎣11100λ−10μ−11⎤⎥⎦
R3↔R2 ⎡⎢⎣1110μ−1100λ−1⎤⎥⎦
Let’s convert C into echelon form
C = ⎡⎢⎣111211λ11μ23⎤⎥⎦
R2→R2–R1, R3→R3–R1
⎡⎢⎣111200λ−1−10μ−111⎤⎥⎦
R2↔R3 ⎡⎢⎣11120μ−11100λ−1−1⎤⎥⎦
When, μ=1, λ=1
Rank of A = 2, Rank of C = 3
→ No solutions possible
When, μ=2, λ=3
Rank of A = 3, Rank of C = 3 = Number of unknowns
→ unique solution
When, μ=1, λ=0
Rank of A = 2, Rank of C = 2
2 < Number of unknowns
∴ Infinitie number of solutions.