The sum of distinct values of $\lambda$ for which the
system of equations
$$\left( {\lambda - 1} \right)x + \left( {3\lambda + 1} \right)y + 2\lambda z = 0$$
$$\left( {\lambda - 1} \right)x + \left( {4\lambda - 2} \right)y + \left( {\lambda + 3} \right)z = 0$$
$2x + \left( {3\lambda + 1} \right)y + 3\left( {\lambda - 1} \right)z = 0$
has non-zero solutions, is ________ .
Answer (integer)
3
Solution
$$\left| {\matrix{
{\lambda - 1} & {3\lambda + 1} & {2\lambda } \cr
{\lambda - 1} & {4\lambda - 2} & {\lambda + 3} \cr
2 & {3\lambda + 1} & {3\left( {\lambda - 1} \right)} \cr
} } \right|$$ = 0
<br><br>R<sub>2</sub> $\to$ R<sub>2</sub>
– R<sub>1</sub>
<br>R<sub>3</sub> $\to$ R<sub>3</sub>
– R<sub>1</sub>
<br><br>$$\left| {\matrix{
{\lambda - 1} & {3\lambda + 1} & {2\lambda } \cr
0 & {\lambda - 3} & { - \lambda + 3} \cr
{3 - \lambda } & 0 & {\lambda - 3} \cr
} } \right| = 0$$
<br><br>C<sub>1</sub> $\to$ C<sub>1</sub>
+ C<sub>3</sub>
<br><br>$$\left| {\matrix{
{3\lambda - 1} & {3\lambda + 1} & {2\lambda } \cr
{ - \lambda + 3} & {\lambda - 3} & { - \lambda + 3} \cr
0 & 0 & {\lambda - 3} \cr
} } \right| = 0$$
<br>$\Rightarrow$ ($\lambda$ - 3) [(3$\lambda$ - 1) ($\lambda$ - 3) – (3 – $\lambda$) (3$\lambda$ + 1)] = 0
<br>$\Rightarrow$ ($\lambda$ – 3) [3$\lambda$<sup>2</sup>
– 10$\lambda$ + 3 –(8$\lambda$ –3$\lambda$<sup>2</sup>
+ 3)] = 0
<br>$\Rightarrow$ ($\lambda$ – 3) (6$\lambda$<sup>2</sup>
– 18$\lambda$) = 0
<br>$\Rightarrow$ (6$\lambda$) ($\lambda$ – 3)<sup>2</sup> = 0
<br>$\Rightarrow$ $\lambda$ = 0, 3
<br>$\therefore$ sum of values of $\lambda$ = 0 + 3 = 3
About this question
Subject: Mathematics · Chapter: Matrices and Determinants · Topic: Types of Matrices and Operations
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