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Joint Entrance Examination
Mathematics
Matrices and Determinants
Question 369
Question 369 of 472
Medium
MCQ
Let \[\lambda \in \] R . The system of linear equations 2x1 - 4x2 + \[\lambda \]x3 = 1 x1 - 6x2 + x3 = 2 \[\lambda \]x1 - 10x2 + 4x3 = 3 is inconsistent for:
A
exactly one positive value of \[\lambda \]
B
exactly one negative value of \[\lambda \]
C
exactly two values of \[\lambda \]
D
every value of \[\lambda \]
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Official solution
D = \[\left| {
(
2
−
4
λ
1
−
6
1
λ
−
10
4
)
\begin{pmatrix} 2 & { - 4} & \lambda \\ 1 & { - 6} & 1 \\ \lambda & { - 10} & 4 \\ \end{pmatrix}
2
1
λ
−
4
−
6
−
10
λ
1
4
} \right|\] = 0 \[ \Rightarrow \] \[\lambda \] = 3, \[ - {2 \over 3}\] D1 = \[\left| {
(
1
−
4
λ
2
−
6
1
3
−
10
4
)
\begin{pmatrix} 1 & { - 4} & \lambda \\ 2 & { - 6} & 1 \\ 3 & { - 10} & 4 \\ \end{pmatrix}
1
2
3
−
4
−
6
−
10
λ
1
4
} \right|\] = 14 + 4(5) + \[\lambda (--2) = --2\[\lambda \] + 6 D2 = \[\left| {
(
2
1
λ
1
2
1
λ
3
4
)
\begin{pmatrix} 2 & 1 & \lambda \\ 1 & 2 & 1 \\ \lambda & 3 & 4 \\ \end{pmatrix}
2
1
λ
1
2
3
λ
1
4
} \right|\] = --2(\[\lambda \] -- 3)(\[\lambda \] + 1) D3 = \[\left| {
(
2
−
4
1
1
−
6
2
λ
−
10
3
)
\begin{pmatrix} 2 & { - 4} & 1 \\ 1 & { - 6} & 2 \\ \lambda & { - 10} & 3 \\ \end{pmatrix}
2
1
λ
−
4
−
6
−
10
1
2
3
} \right|\] = -- 2\[\lambda \] + 6 When , \[\lambda \] = 3 then D = D1 = D2 = D3 = 0 \[ \Rightarrow \] Infinite many solution When \[\lambda \] = \[ - {2 \over 3}\] then D1, D2, D3 none of them is zero so equations are inconsistant. \[ \therefore \] \[\lambda \] = \[ - {2 \over 3}\]
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