Let [, R].

Question

Let $$\lambda, \mu \in \mathbf{R}$$. If the system of equations

$$\begin{aligned}
& 3 x+5 y+\lambda z=3 \
& 7 x+11 y-9 z=2 \
& 97 x+155 y-189 z=\mu
\end{aligned}$$

has infinitely many solutions, then $$\mu+2 \lambda$$ is equal to :

Accepted Answer

$$\begin{aligned}
& 3 x+5 y+\lambda z=3 \
& 7 x+11 y-9 z=2 \
& 97 x+155 y-189 z=\mu
\end{aligned}$$

$$\begin{aligned}
& {\left[\[\begin{array}{ccc} 3 & 5 & \lambda \ 7 & 11 & -9 \ 97 & 155 & -189 \end{array}\end{aligned}$$\right]\left[$$\begin{gathered} x \ y \ z \end{gathered}$$\right]=\left[$$\begin{gathered} 3 \ 2 \ \mu \end{gathered}$$\right]} \
\& A X=B \
\& X=A\^{-1} B \
\& X=\frac{\operatorname{adj} A}{|A|} B
\]

For Infinitely many solution

$$\begin{aligned}
& |A|=0 \text { and }(\operatorname{adj} A) B=0 \
& \left|\[\begin{array}{ccc} 3 & 5 & \lambda \ 7 & 11 & -9 \ 97 & 155 & -189 \end{array}\end{aligned}$$\right|=0 \
\& -2052+2250+18 \lambda=0 \
\& \Rightarrow \lambda=-11 \
\& \operatorname{adj} A=\left[$$\begin{array}{ccc} -684 & -760 & 76 \ 450 & 500 & -50 \ 18 & 20 & -2 \end{array}$$\right] \
\& (\operatorname{adj} A) B=\left[$$\begin{array}{ccc} 684 & -760 & 76 \ 450 & 500 & -50 \ 18 & 20 & -2 \end{array}$$\right]\left[$$\begin{gathered} 3 \ 2 \ \mu \end{gathered}$$\right]=\left[$$\begin{gathered} 0 \ 0 \ 0 \end{gathered}$$\right] \
\& \Rightarrow 54+40-2 \mu=0 \
\& \Rightarrow 2 \mu=94 \
\& \Rightarrow \mu=47 \
\& \Rightarrow \mu+2 \lambda=25
\]

Solutions · 1