Practicing Success
The values of $\lambda$ for which the following system of equations has unique solution. $\lambda x+3 y-z=1$ $x+2 y+z=2$ $-\lambda x+y+2 z=-1$ are |
$\lambda \neq \frac{5}{2}$ $\lambda \neq \frac{3}{2}$ $\lambda \neq \frac{7}{2}$ $\lambda \neq \frac{-7}{2}$ |
$\lambda \neq \frac{-7}{2}$ |
$\lambda x+3 y-z=1$ ......(1) $x+2 y+z=2$ ......(2) $-\lambda x+y+2 z=-1$ ......(3) so $A=\left[\begin{array}{ccc}\lambda & 3 & -1 \\ 1 & 2 & 1 \\ -\lambda & 1 & 2\end{array}\right]$ for unique solution $|A| \neq 0$ so $|A|=\left|\begin{array}{ccc}\lambda & 3 & -1 \\ 1 & 2 & 1 \\ -\lambda & 1 & 2\end{array}\right|$ so $|A|=\lambda\left|\begin{array}{ll}2 & 1 \\ 1 & 2\end{array}\right|-3\left|\begin{array}{cc}1 & 1 \\ -\lambda & 2\end{array}\right|-1\left|\begin{array}{cc}1 & 2 \\ -\lambda & 1\end{array}\right|$ upon solving $|A|=-2 \lambda-7 \neq 0$ $\Rightarrow 2 \lambda \neq-7$ $\lambda \neq \frac{-7}{2}$ |