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Joint Entrance Examination
Mathematics
Matrices and Determinants
Question 431
Question 431 of 472
Medium
MCQ
Consider the following system of equations \[\alpha x+2y+z=1\] \[2\alpha x+3y+z=1\] \[3x+\alpha y+2z=\beta\] for some \[\alpha,\beta\in \mathbb{R}\]. Then which of the following is NOT correct.
A
It has a solution for all \[\alpha\ne-1\] and \[\beta=2\]
B
It has no solution if \[\alpha=-1\] and \[\beta\ne2\]
C
It has no solution for \[\alpha=-1\] and for all \[\beta \in \mathbb{R}\]
D
It has no solution for \[\alpha=3\] and for all \[\beta\ne2\]
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Official solution
D
=
∣
α
2
1
2
α
3
1
3
α
2
∣
=
0
⇒
α
=
−
1
,
3
D=\left|\begin{array}{|ccccccc|}\alpha & 2 & 1 \\ 2 \alpha & 3 & 1 \\ 3 & \alpha & 2\end{array}\right|=0 \Rightarrow \alpha=-1,3
D
=
α
2
α
3
2
3
α
1
1
2
=
0
⇒
α
=
−
1
,
3
D
x
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∣
2
1
1
3
1
1
α
2
β
∣
=
0
⇒
β
=
2
D_{x}=\left|\begin{array}{|ccccccc|}2 & 1 & 1 \\ 3 & 1 & 1 \\ \alpha & 2 & \beta\end{array}\right|=0 \Rightarrow \beta=2
D
x
=
2
3
α
1
1
2
1
1
β
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0
⇒
β
=
2
D
y
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∣
α
1
1
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α
1
1
3
2
β
∣
=
0
D_{y}=\left|\begin{array}{|ccccccc|}\alpha & 1 & 1 \\ 2 \alpha & 1 & 1 \\ 3 & 2 & \beta\end{array}\right|=0
D
y
=
α
2
α
3
1
1
2
1
1
β
=
0
D
z
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∣
α
2
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1
3
α
β
∣
=
0
D_{z}=\left|\begin{array}{|ccccccc|}\alpha & 2 & 1 \\ 2 \alpha & 3 & 1 \\ 3 & \alpha & \beta\end{array}\right|=0
D
z
=
α
2
α
3
2
3
α
1
1
β
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0
β
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2
,
α
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−
1
\beta=2, \alpha=-1
β
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2
,
α
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−
1
α
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−
1
,
β
=
2
\alpha=-1, \beta=2
α
=
−
1
,
β
=
2
Infinite solution
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