Consider this equations system of the form $Ax=b$ where
\begin{align*} A=\begin{pmatrix} 1 & 2 & 0 & 3\\ 2 & 1 & 0 & \alpha\\ 0 & 1 & 1 & 0\\ 0 & 0 & 1 & 3 \end{pmatrix} \ \ \ \ \ \ \ \ \ \ \ \ \ \ b=\begin{pmatrix} 0\\ \beta\\ 1\\ 0 \end{pmatrix} \end{align*}
1. For which values $\alpha$ and $\beta$ the system has solution?
2. For which values $\alpha$ and $\beta$ the system has an unique solution?
3. If the system doesn't have an unique solution, describe all the solutions
- What I did for 2 is to calculate:
\begin{align*} Det(A)=-15+\alpha \end{align*} So my answer is that the system of equations has an unique solution $\iff$ $A^{-1}$ exists $\iff$ $\alpha\neq15$
i.e., for $\alpha\neq15$ and $\forall \beta$ the system of equations has an unique solution. Am I correct?
- For 3 what I did is this:
\begin{align*} x_1+2x_2+3x_4&=0\\ 2x_1+x_2+\alpha x_4&=\beta\\ x_2+x_3&=1\\ x_3+x_4&=0 \end{align*}
From that, I got that the solutions has this form:
\begin{align*} \begin{pmatrix} x_1\\ x_2\\ x_3\\ x_4 \end{pmatrix}=\begin{pmatrix} -2(2x_4+1)\\ 1+x_4\\ -x_4\\ x_4 \end{pmatrix} \ \ \ \ \ \ \ \ \ \text{with } x_4=\frac{3+\beta}{\alpha-9} \end{align*}
- My doubts:
a) Am I correct in 2?
b) In 3, I got $x_4=\frac{3+\beta}{\alpha-9}$, but what happens if $\alpha=9$? Does that means that with $\alpha=9$ then the system hasn't solution?
c) For 1, how can I determine if the system has solution?
I would really appreciate your help!
