Let be $A$ and $B$, $n \times n$ matrices. And let be a system of equations $Ax=b$.
\begin{align*} A=\begin{pmatrix} 1 & 2 & 0 & 3\\ 2 & 1 & 0 & 15\\
0 & 1 & 1 & 0\\ 0 & 0 & 1 & 3 \end{pmatrix} \ \ \ \ \ \ \ \ \ \ \ \ \
\ b=\begin{pmatrix} 0\\ \beta\\ 1\\ 0 \end{pmatrix} \end{align*}
Find the $\beta$ values such that the system has no solution.
I calculated $\det(A)=0$ , this means that the system has not a unique solution. (So the system can have infinite solutions or can have no solution). Also I know that the rank of $A$ is 3. So, how can I determine the $\beta$ values such that this system hasn't solutions?
Best Answer
Hint: Just use the gaussian row elimination method to analyze the system of linear equations and for which $\beta$ values have no solution. $$\left(\begin{array}{cccc|c}1&2&0&3&0\\2&1&0&15&\beta\\0&1&1&0&1\\0&0&1&3&0\end{array}\right)\sim\cdots \sim \left(\begin{array}{cccc|c}1&2&0&3&0\\0&-3&0&\color{blue}{9}&\beta\\0&0&1&\color{blue}{3}&\frac{\beta+3}{3}\\0&0&0&0&\color{blue}{\beta+3}\end{array}\right)$$