Linear Algebra Complex A homogonous linear system is given by: > x1 + x2 = 0 > > a · x2 + x3 = 0 > > 2·x1 + x2 + a·x3 =0 where a ∈ C, a) Find the determinant of A and give the values of a for which matrix A is regular. b) Determine for any value of 'a' trap matrix (T) by means of row operations. c) Provide for any value of 'a' all complex solutions of the equation system. I have solved a) and I got that a≠ ±i, but I am stuck with b) and c). For b), i use the the row operations and I got that a^2 + 1 = 0, and thus a=±i. Any help would be great.

Ok some start along the way.

First if we order $x_1,x_2,x_3$ as columns we can write the equation system as a matrix:

$$\left[\begin{array}{rrr|r}1&1&0&0\\\0&a&1&0\\\2&1&a&0\end{array}\right]$$

where the first row means $x_1+x_2 = 0$ and so on...

If you replace row 3 with 2 times row 1 minus row 3

$$\left[\begin{array}{rrr|r}1&1&0&0\\\0&a&1&0\\\0&-1&a&0\end{array}\right]$$

Now the tricky part is the lower right $2\times 2$ block, but you find that $a = \pm i$ is special since (i,1) is parallell to $(-1,i)$ which you can realize by multiplying one by $-i$ or the other by $i$.

So unless $a = \pm i$ we can be sure that the solution can only be the 0 vector because for any non-degenerate homogenous equation system only has the zero vectors as solution.

But if $a = \pm i$ we would have to solve it by paramterization, let us assume $x_1 = t$, then by first equation $x_2 = -t$ and then we can pick any of eq 2 or 3 to express $x_3$ in $t$.