Consider the system of equations x+y+z=6,x+2y+3z=10,x+2y+λz=μ
The system has infinite solutions if:
A
λ≠3
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B
λ=3,μ=10
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C
λ=3,μ≠10
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D
none of these
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Solution
The correct option is Bλ=3,μ=10 Given system of equations AX=B where A=⎡⎢⎣11112312λ⎤⎥⎦, X=⎡⎢⎣xyz⎤⎥⎦,B=⎡⎢⎣610μ⎤⎥⎦ For infinitely many solution, D=0 ⇒∣∣
∣∣11112312λ∣∣
∣∣=0 ⇒2(λ−3)−(λ−3)=0 ⇒λ=3 Also, D1=∣∣
∣∣6111023μ23∣∣
∣∣=0 ⇒μ=10 D2=∣∣
∣∣16111031μ3∣∣
∣∣=0 ⇒μ=10 D3=∣∣
∣∣116121012μ∣∣
∣∣=0 ⇒μ=10