How do I find all values of: $$ a \in R $$ for the system of equations below?
$$ \left\{ \begin{array}{c} x_2-3x_3+2x_4=1 \\ x_1+3x_2-x_3-x_4=9 \\ x_1+4x_2-4x_3+(7+a)x_4=16+a \\ x_1-2x_2+14x_3-11x_4=4 \end{array} \right. $$
I'm not asking for a straight answer to the solution, but more of a method for finding it :-)
Thank you
The matrix of coefficients is: $\begin{pmatrix} 0 & 1 & -3 & 2 \\ 1 & 3 & -1 & -1 \\ 1 & 4 & -4 & 7+a \\ 1 & -2 & 14 & -11 \end{pmatrix}$ Evaluate the determinant (A) of this matrix as a function of $a$. All values of $a$ are allowed except when A$=0$.
In fact, the expression for A will be linear in $a$, so there is only one value of $a$ that is not allowed.