what value of K does the system have a unique solution

If $k\neq -\frac12$ $$\left( \begin{matrix} 1 & k & -1 & | & 2\\ 2 & -1 & k & | & 5 \\ 1 & 10 & -6 & | & 1 \\ \end{matrix} \right) \xrightarrow[\text{$R_3=R_3-R_1$}]{\text{$R_2=R_2-2R_1$}}$$ $$\left( \begin{matrix} 1 & k & -1 & | & 2\\ 0 & -1-2k & k+2 & | & 1 \\ 0 & 10-k & -5 & | & -1 \\ \end{matrix} \right) \xrightarrow[\text{$(2k+1\neq0)$}]{R_3=R_3-(\frac{10-k}{2k+1})R_2} $$ $$\left( \begin{matrix} 1 & k & -1 & | & 2\\ 0 & -1-2k & k+2 & | & 1 \\ 0 & 0 & \frac{-k^2-2k+15}{2k+1} & | & \frac{-3k+9}{2k+1} \\ \end{matrix} \right) $$ Which has a unique solution $\iff \frac{-3k+9}{2k+1} \neq 0 \iff -3k+9\neq 0 \iff k\neq3$

If $k=-\frac12$ it is easy to show that the system has a unique solution.

Tags:

Linear Algebra

Related

find the remainder for $\color{purple}{TETRATION}$ such that $( {^{2021}2021}(\mod13))$ Analyse convergence of: $\displaystyle \sum^{\infty}_{n=3} \left (\sqrt[3]{n^3 + 2} - \sqrt{n^2 + 7}\right)$ Evaluate $\int_{0}^{1} \frac{\ln(1 + x + x^2 + \ldots + x^n)}{x}\mathrm d x$ The converse problem about duality of $L^p$ Cartesian product of two collections of sets? Prove this formula $\frac{1-r\cos(x)}{1-2r\cos(x)+r^2}= 1+\sum_{n=1}^{+\infty}r^{n}\cos\left(nx\right)$ The formulas of prostapheresis: memorization technique Evaluating $I(z,s)=\int_0^1\int_0^1\left(1-\frac{(1-x)(1-y)}{(1-(1-z)x)(1-(1-z)y)}\right)^{s-2}\,\mathrm dx\mathrm dy$ Is Bayes' Theorem really that interesting? Summing $1+\cos(\theta)+\cos(2\theta) +\cdots + \cos(n\theta)$ Integration of $\int_0^{\infty}\frac{e^{(\lambda +is)t}-e^{-(\lambda +is)t}}{e^{\pi t}-e^{-\pi t}}dt$ Strategy of a game

Recent Posts