Question

Consider the system of equations

$x+2y+3z=6,$

$4x+5y+6z=\lambda ,$

$7x+8y+9z=24.$

Then, the value of $\lambda $ for which the system has infinite solutions is

• A. $15$
• B. $20$
• C. $16$
• D. $30$

Answer 1

Answer: (A)

Note that $D=\begin{vmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{vmatrix}=0\left(R_{3} \rightarrow R_{3} - 2 R_{2} + R_{1}\right)$
$D_{1}=\begin{vmatrix} 6 & 2 & 3 \\ \lambda & 5 & 6 \\ 24 & 8 & 9 \end{vmatrix}=-3\left(30 - 2 \lambda \right)$
$D_{2}=\begin{vmatrix} 1 & 6 & 3 \\ 4 & \lambda & 6 \\ 7 & 24 & 9 \end{vmatrix}=6\left(30 - 2 \lambda \right)$
$D_{3}=\begin{vmatrix} 1 & 2 & 6 \\ 4 & 5 & \lambda \\ 7 & 8 & 24 \end{vmatrix}=-3\left(30 - 2 \lambda \right)$
Hence, for $\lambda =15$ the system has infinite solutions