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:
Solution
D = $$\left| {\matrix{
2 & { - 4} & \lambda \cr
1 & { - 6} & 1 \cr
\lambda & { - 10} & 4 \cr
} } \right|$$ = 0
<br><br>$\Rightarrow$ $\lambda$ = 3, $- {2 \over 3}$
<br><br>D<sub>1</sub> = $$\left| {\matrix{
1 & { - 4} & \lambda \cr
2 & { - 6} & 1 \cr
3 & { - 10} & 4 \cr
} } \right|$$
<br><br>= 14 + 4(5) + $\lambda$(–2)
<br><br>= –2$\lambda$ + 6
<br><br>D<sub>2</sub> = $$\left| {\matrix{
2 & 1 & \lambda \cr
1 & 2 & 1 \cr
\lambda & 3 & 4 \cr
} } \right|$$
<br><br>= –2($\lambda$ – 3)($\lambda$ + 1)
<br><br>D<sub>3</sub> = $$\left| {\matrix{
2 & { - 4} & 1 \cr
1 & { - 6} & 2 \cr
\lambda & { - 10} & 3 \cr
} } \right|$$
<br><br>= – 2$\lambda$ + 6
<br><br>When , $\lambda$ = 3 then
<br><br>D = D<sub>1</sub> = D<sub>2</sub> = D<sub>3</sub> = 0
<br><br>$\Rightarrow$ Infinite many solution
<br><br>When $\lambda$ = $- {2 \over 3}$ then D<sub>1</sub>, D<sub>2</sub>, D<sub>3</sub> none of them is zero so equations are inconsistant.
<br><br>$\therefore$ $\lambda$ = $- {2 \over 3}$
About this question
Subject: Mathematics · Chapter: Matrices and Determinants · Topic: Types of Matrices and Operations
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