Let , R.

Question

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

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

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

Accepted Answer

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

$\begin{gathered} [[|\begin{array}{ccc}3 \quad  5 \quad  \lambda \ 7 \quad  11 \quad  -9 \ 97 \quad  155 \quad  -189\end{array}]]|[[|\begin{array}{cc}x \ y \ z\end{array}]]|=[[|\begin{array}{cc}3 \ 2 \ \mu \end{array}]]| \ AX=B \ X={A}^{-1}B \ X=\frac{adjA}{|A|}B \end{gathered}$

For Infinitely many solution

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

Solutions · 1