Question

For the system of linear equations $\alpha x+y+z=1, x+\alpha y+z=1, x+y+\alpha z=\beta$, which one of the following statements is NOT correct ?

• A. It has infinitely many solutions if $\alpha=2$ and $\beta=-1$
• B. $x+y+z=\frac{3}{4}$ if $\alpha=2$ and $\beta=1$
• C. It has infinitely many solutions if $\alpha=1$ and $\beta=1$
• D. It has no solution if $\alpha=-2$ and $\beta=1$

Answer 1

Answer: (A)

It has infinitely many solutions if $\alpha=2$ and $\beta=-1$

Answer 2

∣∣​α11​1α1​11α​∣∣​=0
α(α2−1)−1(α−1)+1(1−α)=0
α3−3α+2=0
α2(α−1)+α(α−1)−2(α−1)=0
(α−1)(α2+α−2)=0
α=1,α=−2,1
For α=1,β=1

Δ1​=∣∣​111​121​112​∣∣​=3−1−1⇒x=41​
Δ2​=∣∣​211​111​112​∣∣​=2−1=1⇒y=41​
Δ3​=∣∣​211​121​111​∣∣​=2−1=1⇒z=41​
For α=2⇒ unique solution