Linear System - Laplace - Determinant

Question

Can somebody help me? I need to find the determinant of the related matrix with Laplace's method. What is the easiest way to find it?

$x+y-z+w=1\\ x+2y+z-w=-1\\ y+2z-2w=-2\\ kx+3z=0$

Thank you for your help!

Answer 1

I suppose you are asking about the determinant of the linear system, which you wrote in your question, i.e., $$|A|= \begin{vmatrix} 1 & 1 &-1 & 1 \\ 1 & 2 & 1 &-1 \\ 0 & 1 & 2 &-2 \\ k & 0 & 3 & 0 \end{vmatrix} $$

If you use Laplace expansion w.r.t. the fourth row you get: $$|A|= -k \begin{vmatrix} 1 &-1 & 1 \\ 2 & 1 &-1 \\ 1 & 2 &-2 \\ \end{vmatrix} -3 \begin{vmatrix} 1 & 1 & 1 \\ 1 & 2 &-1 \\ 0 & 1 &-2 \\ \end{vmatrix} $$ The you have to calculate these two $3\times3$ determinants. (I suppose this is what you were trying to do in your comment. From what you wrote there is seems that you had the number $3$ in the second column of your matrix.)

You can then check the results online, for example, in WolframAlpha.

However, if you notice that the second row of the matrix $A$ is exactly the sum of the first row and the third row, then you know that the rows are linearly dependent and thus the determinant is zero. (But you should get the same result using your method.)