Q55 / 125 · BITSAT 2016 · Mathematics

If the matrix $\left[\begin{array}{ccc}1 & 3 & \lambda+2 \\ 2 & 4 & 8 \\ 3 & 5 & 10\end{array}\right]$ is singular, then $\lambda=$

  • A
    -2
  • Correct answer
    B
    4
  • C
    2
  • D
    -4

Solution

$|\mathrm{A}|=0$ as the matrix $\mathrm{A}$ is singular $$ \therefore|\mathrm{A}|=\left|\begin{array}{ccc} 1 & 3 & \lambda+2 \\ 2 & 4 & 8 \\ 3 & 5 & 10 \end{array}\right|=0 $$ Apply $\mathrm{R}_{2} \rightarrow \mathrm{R}_{2}-2 \mathrm{R}_{1}$ and $\mathrm{R}_{3} \rightarrow \mathrm{R}_{3}-3 \mathrm{R}_{1}$ and expand. $-2(4-3 \lambda)+4(4-2 \lambda)=0$ $$ \Rightarrow 8-2 \lambda=0 \Rightarrow \lambda=4 $$ For $\lambda=4,$ the second and the third column are proportional.

Hence, the correct answer is option: 'B'.

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